From CoqCP Require Import Options Imperative DisjointSetUnion ListsEqual ExistsInRange SwapUpdate.
[Loading ML file ring_plugin.cmxs (using legacy method) ... done]
[Loading ML file zify_plugin.cmxs (using legacy method) ... done]
[Loading ML file micromega_plugin.cmxs (using legacy method) ... done]
[Loading ML file btauto_plugin.cmxs (using legacy method) ... done]
[Loading ML file ssrmatching_plugin.cmxs (using legacy method) ... done]
[Loading ML file ssreflect_plugin.cmxs (using legacy method) ... done]
From Generated Require Import DisjointSetUnion.
From stdpp Require Import numbers list.
Require Import Coq.Logic.FunctionalExtensionality.
Require Import Wellfounded.

Definition state : BlockchainState := fun address =>
  if decide (address = repeat (0%Z) 20) then
    BlockchainContract arrayIndex0 _ (arrayType _ environment0) (arrays _ environment0) 100000%Z (funcdef_0__main (fun x => false) (fun x => 0%Z) (fun x => repeat 0%Z 20))
  else ExternallyOwned 0%Z.

Definition stateAfterInteractions arrays money : BlockchainState := fun address =>
  if decide (address = repeat (0%Z) 20) then
    BlockchainContract arrayIndex0 _ (arrayType _ environment0) arrays (100000%Z - money) (funcdef_0__main (fun x => false) (fun x => 0%Z) (fun x => repeat 0%Z 20))
  else if decide (address = repeat (1%Z) 20) then ExternallyOwned money else ExternallyOwned 0%Z.

Fixpoint interact (state : BlockchainState) (interactions : list (Z * Z)) :=
  match interactions with
  | [] => getBalance (state (repeat 1%Z 20))
  | (a, b) :: tail =>
    match invokeContract (repeat 1%Z 20) (repeat 0%Z 20) 0%Z state state [a; b] 1 with
    | Some (_, changedState) => interact changedState tail
    | None => 0%Z
    end
  end.

Definition optimalInteractions (x : list (Z * Z)) := forall x', Z.le (interact state x') (interact state x).

Inductive Slot :=
| ReferTo (x : nat)
| Ancestor (x : Tree).

Definition noIllegalIndices (dsu : list Slot) := forall x y, nth x dsu (Ancestor Unit) = ReferTo y -> y < length dsu.

Fixpoint convertToArray (x : list Slot) : list Z :=
  match x with
  | [] => []
  | (ReferTo x) :: tail => (Z.of_nat x) :: convertToArray tail
  | (Ancestor x) :: tail => (Z.sub 256%Z (Z.of_nat (leafCount x))) :: convertToArray tail
  end.

Lemma nthConvert (x : list Slot) (n : nat) (hN : n < length x) : nth n (convertToArray x) (0%Z) = match nth n x (Ancestor Unit) with | ReferTo c => Z.of_nat c | Ancestor c => Z.sub 256%Z (Z.of_nat (leafCount c)) end.
x: list Slot
n: nat
hN: n < length x
nth n (convertToArray x) 0%Z =
match nth n x (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end
Proof.
x: list Slot
n: nat
hN: n < length x
nth n (convertToArray x) 0%Z =
match nth n x (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end
  revert n hN.
x: list Slot
∀ _0 : nat,
  _0 < length x
  → nth _0 (convertToArray x) 0%Z =
    match nth _0 x (Ancestor Unit) with
    | ReferTo _1 => Z.of_nat _1
    | Ancestor _1 => (256 - Z.of_nat (leafCount _1))%Z
    end induction x as [| head tail IH]; intros n hN.
n: nat
hN: n < length []
nth n (convertToArray []) 0%Z =
match nth n [] (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end
head: Slot
tail: list Slot
IH: ∀ _0 : nat,
  _0 < length tail
  → nth _0 (convertToArray tail) 0%Z =
    match nth _0 tail (Ancestor Unit) with
    | ReferTo _1 => Z.of_nat _1
    | Ancestor _1 =>
        (256 - Z.of_nat (leafCount _1))%Z
    end
n: nat
hN: n < length (head :: tail)
nth n (convertToArray (head :: tail)) 0%Z =
match nth n (head :: tail) (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end
  {
n: nat
hN: n < length []
nth n (convertToArray []) 0%Z =
match nth n [] (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end simpl in hN.
n: nat
hN: n < 0
nth n (convertToArray []) 0%Z =
match nth n [] (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end lia. }
head: Slot
tail: list Slot
IH: ∀ _0 : nat,
  _0 < length tail
  → nth _0 (convertToArray tail) 0%Z =
    match nth _0 tail (Ancestor Unit) with
    | ReferTo _1 => Z.of_nat _1
    | Ancestor _1 =>
        (256 - Z.of_nat (leafCount _1))%Z
    end
n: nat
hN: n < length (head :: tail)
nth n (convertToArray (head :: tail)) 0%Z =
match nth n (head :: tail) (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end
  destruct n as [| n].
head: Slot
tail: list Slot
IH: ∀ _0 : nat,
  _0 < length tail
  → nth _0 (convertToArray tail) 0%Z =
    match nth _0 tail (Ancestor Unit) with
    | ReferTo _1 => Z.of_nat _1
    | Ancestor _1 =>
        (256 - Z.of_nat (leafCount _1))%Z
    end
hN: 0 < length (head :: tail)
nth 0 (convertToArray (head :: tail)) 0%Z =
match nth 0 (head :: tail) (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end
head: Slot
tail: list Slot
IH: ∀ _0 : nat,
  _0 < length tail
  → nth _0 (convertToArray tail) 0%Z =
    match nth _0 tail (Ancestor Unit) with
    | ReferTo _1 => Z.of_nat _1
    | Ancestor _1 =>
        (256 - Z.of_nat (leafCount _1))%Z
    end
n: nat
hN: S n < length (head :: tail)
nth (S n) (convertToArray (head :: tail)) 0%Z =
match nth (S n) (head :: tail) (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end
  -
head: Slot
tail: list Slot
IH: ∀ _0 : nat,
  _0 < length tail
  → nth _0 (convertToArray tail) 0%Z =
    match nth _0 tail (Ancestor Unit) with
    | ReferTo _1 => Z.of_nat _1
    | Ancestor _1 =>
        (256 - Z.of_nat (leafCount _1))%Z
    end
hN: 0 < length (head :: tail)
nth 0 (convertToArray (head :: tail)) 0%Z =
match nth 0 (head :: tail) (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end simpl.
head: Slot
tail: list Slot
IH: ∀ _0 : nat,
  _0 < length tail
  → nth _0 (convertToArray tail) 0%Z =
    match nth _0 tail (Ancestor Unit) with
    | ReferTo _1 => Z.of_nat _1
    | Ancestor _1 =>
        (256 - Z.of_nat (leafCount _1))%Z
    end
hN: 0 < length (head :: tail)
nth 0
  match head with
  | ReferTo _0 => Z.of_nat _0 :: convertToArray tail
  | Ancestor _0 =>
      (256 - Z.of_nat (leafCount _0))%Z
      :: convertToArray tail
  end 0%Z =
match head with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end destruct head; reflexivity.
  -
head: Slot
tail: list Slot
IH: ∀ _0 : nat,
  _0 < length tail
  → nth _0 (convertToArray tail) 0%Z =
    match nth _0 tail (Ancestor Unit) with
    | ReferTo _1 => Z.of_nat _1
    | Ancestor _1 =>
        (256 - Z.of_nat (leafCount _1))%Z
    end
n: nat
hN: S n < length (head :: tail)
nth (S n) (convertToArray (head :: tail)) 0%Z =
match nth (S n) (head :: tail) (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end rewrite -> (ltac:(destruct head; reflexivity) : nth (S n) (convertToArray (head :: tail)) 0%Z = nth n (convertToArray tail) 0%Z), (ltac:(simpl; reflexivity) : nth (S _) (_ :: _) _ = _).
head: Slot
tail: list Slot
IH: ∀ _0 : nat,
  _0 < length tail
  → nth _0 (convertToArray tail) 0%Z =
    match nth _0 tail (Ancestor Unit) with
    | ReferTo _1 => Z.of_nat _1
    | Ancestor _1 =>
        (256 - Z.of_nat (leafCount _1))%Z
    end
n: nat
hN: S n < length (head :: tail)
nth n (convertToArray tail) 0%Z =
match nth n tail (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end apply IH.
head: Slot
tail: list Slot
IH: ∀ _0 : nat,
  _0 < length tail
  → nth _0 (convertToArray tail) 0%Z =
    match nth _0 tail (Ancestor Unit) with
    | ReferTo _1 => Z.of_nat _1
    | Ancestor _1 =>
        (256 - Z.of_nat (leafCount _1))%Z
    end
n: nat
hN: S n < length (head :: tail)
n < length tail simpl in hN.
head: Slot
tail: list Slot
IH: ∀ _0 : nat,
  _0 < length tail
  → nth _0 (convertToArray tail) 0%Z =
    match nth _0 tail (Ancestor Unit) with
    | ReferTo _1 => Z.of_nat _1
    | Ancestor _1 =>
        (256 - Z.of_nat (leafCount _1))%Z
    end
n: nat
hN: S n < S (length tail)
n < length tail lia.
Qed.

Lemma insertConvertReferTo (x : list Slot) (n : nat) (hN : n < length x) (v : Z) (hU : Z.le 0 v) : <[n := v]> (convertToArray x) = convertToArray (<[n := ReferTo (Z.to_nat v)]> x).
x: list Slot
n: nat
hN: n < length x
v: Z
hU: (0 ≤ v)%Z
<[n:=v]> (convertToArray x) =
convertToArray (<[n:=ReferTo (Z.to_nat v)]> x)
Proof.
x: list Slot
n: nat
hN: n < length x
v: Z
hU: (0 ≤ v)%Z
<[n:=v]> (convertToArray x) =
convertToArray (<[n:=ReferTo (Z.to_nat v)]> x)
  revert n hN.
x: list Slot
v: Z
hU: (0 ≤ v)%Z
∀ _0 : nat,
  _0 < length x
  → <[_0:=v]> (convertToArray x) =
    convertToArray (<[_0:=ReferTo (Z.to_nat v)]> x) induction x as [| head tail IH]; intros n hN.
v: Z
hU: (0 ≤ v)%Z
n: nat
hN: n < length []
<[n:=v]> (convertToArray []) =
convertToArray (<[n:=ReferTo (Z.to_nat v)]> [])
head: Slot
tail: list Slot
v: Z
hU: (0 ≤ v)%Z
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=v]> (convertToArray tail) =
    convertToArray
      (<[_0:=ReferTo (Z.to_nat v)]> tail)
n: nat
hN: n < length (head :: tail)
<[n:=v]> (convertToArray (head :: tail)) =
convertToArray
  (<[n:=ReferTo (Z.to_nat v)]> (head :: tail))
  {
v: Z
hU: (0 ≤ v)%Z
n: nat
hN: n < length []
<[n:=v]> (convertToArray []) =
convertToArray (<[n:=ReferTo (Z.to_nat v)]> []) simpl in hN.
v: Z
hU: (0 ≤ v)%Z
n: nat
hN: n < 0
<[n:=v]> (convertToArray []) =
convertToArray (<[n:=ReferTo (Z.to_nat v)]> []) lia. }
head: Slot
tail: list Slot
v: Z
hU: (0 ≤ v)%Z
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=v]> (convertToArray tail) =
    convertToArray
      (<[_0:=ReferTo (Z.to_nat v)]> tail)
n: nat
hN: n < length (head :: tail)
<[n:=v]> (convertToArray (head :: tail)) =
convertToArray
  (<[n:=ReferTo (Z.to_nat v)]> (head :: tail))
  destruct n as [| n].
head: Slot
tail: list Slot
v: Z
hU: (0 ≤ v)%Z
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=v]> (convertToArray tail) =
    convertToArray
      (<[_0:=ReferTo (Z.to_nat v)]> tail)
hN: 0 < length (head :: tail)
<[0:=v]> (convertToArray (head :: tail)) =
convertToArray
  (<[0:=ReferTo (Z.to_nat v)]> (head :: tail))
head: Slot
tail: list Slot
v: Z
hU: (0 ≤ v)%Z
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=v]> (convertToArray tail) =
    convertToArray
      (<[_0:=ReferTo (Z.to_nat v)]> tail)
n: nat
hN: S n < length (head :: tail)
<[S n:=v]> (convertToArray (head :: tail)) =
convertToArray
  (<[S n:=ReferTo (Z.to_nat v)]> (head :: tail))
  -
head: Slot
tail: list Slot
v: Z
hU: (0 ≤ v)%Z
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=v]> (convertToArray tail) =
    convertToArray
      (<[_0:=ReferTo (Z.to_nat v)]> tail)
hN: 0 < length (head :: tail)
<[0:=v]> (convertToArray (head :: tail)) =
convertToArray
  (<[0:=ReferTo (Z.to_nat v)]> (head :: tail)) simpl.
head: Slot
tail: list Slot
v: Z
hU: (0 ≤ v)%Z
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=v]> (convertToArray tail) =
    convertToArray
      (<[_0:=ReferTo (Z.to_nat v)]> tail)
hN: 0 < length (head :: tail)
<[0:=v]>
  match head with
  | ReferTo _0 => Z.of_nat _0 :: convertToArray tail
  | Ancestor _0 =>
      (256 - Z.of_nat (leafCount _0))%Z
      :: convertToArray tail
  end = Z.of_nat (Z.to_nat v) :: convertToArray tail destruct head; simpl; rewrite Z2Nat.id; try lia; reflexivity.
  -
head: Slot
tail: list Slot
v: Z
hU: (0 ≤ v)%Z
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=v]> (convertToArray tail) =
    convertToArray
      (<[_0:=ReferTo (Z.to_nat v)]> tail)
n: nat
hN: S n < length (head :: tail)
<[S n:=v]> (convertToArray (head :: tail)) =
convertToArray
  (<[S n:=ReferTo (Z.to_nat v)]> (head :: tail)) simpl.
head: Slot
tail: list Slot
v: Z
hU: (0 ≤ v)%Z
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=v]> (convertToArray tail) =
    convertToArray
      (<[_0:=ReferTo (Z.to_nat v)]> tail)
n: nat
hN: S n < length (head :: tail)
<[S n:=v]>
  match head with
  | ReferTo _0 => Z.of_nat _0 :: convertToArray tail
  | Ancestor _0 =>
      (256 - Z.of_nat (leafCount _0))%Z
      :: convertToArray tail
  end =
match head with
| ReferTo _0 =>
    Z.of_nat _0
    :: convertToArray
         (<[n:=ReferTo (Z.to_nat v)]> tail)
| Ancestor _0 =>
    (256 - Z.of_nat (leafCount _0))%Z
    :: convertToArray
         (<[n:=ReferTo (Z.to_nat v)]> tail)
end destruct head; simpl; rewrite IH; try reflexivity; simpl in hN; lia.
Qed.

Lemma insertConvertAncestor (x : list Slot) (n : nat) (hN : n < length x) (v : Tree) : <[n := (256 - Z.of_nat (leafCount v))%Z]> (convertToArray x) = convertToArray (<[n := Ancestor v]> x).
x: list Slot
n: nat
hN: n < length x
v: Tree
<[n:=(256 - Z.of_nat (leafCount v))%Z]>
  (convertToArray x) =
convertToArray (<[n:=Ancestor v]> x)
Proof.
x: list Slot
n: nat
hN: n < length x
v: Tree
<[n:=(256 - Z.of_nat (leafCount v))%Z]>
  (convertToArray x) =
convertToArray (<[n:=Ancestor v]> x)
  revert n hN.
x: list Slot
v: Tree
∀ _0 : nat,
  _0 < length x
  → <[_0:=(256 - Z.of_nat (leafCount v))%Z]>
      (convertToArray x) =
    convertToArray (<[_0:=Ancestor v]> x) induction x as [| head tail IH]; intros n hN.
v: Tree
n: nat
hN: n < length []
<[n:=(256 - Z.of_nat (leafCount v))%Z]>
  (convertToArray []) =
convertToArray (<[n:=Ancestor v]> [])
head: Slot
tail: list Slot
v: Tree
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=(256 - Z.of_nat (leafCount v))%Z]>
      (convertToArray tail) =
    convertToArray (<[_0:=Ancestor v]> tail)
n: nat
hN: n < length (head :: tail)
<[n:=(256 - Z.of_nat (leafCount v))%Z]>
  (convertToArray (head :: tail)) =
convertToArray (<[n:=Ancestor v]> (head :: tail)) {
v: Tree
n: nat
hN: n < length []
<[n:=(256 - Z.of_nat (leafCount v))%Z]>
  (convertToArray []) =
convertToArray (<[n:=Ancestor v]> []) simpl in hN.
v: Tree
n: nat
hN: n < 0
<[n:=(256 - Z.of_nat (leafCount v))%Z]>
  (convertToArray []) =
convertToArray (<[n:=Ancestor v]> []) lia. }
head: Slot
tail: list Slot
v: Tree
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=(256 - Z.of_nat (leafCount v))%Z]>
      (convertToArray tail) =
    convertToArray (<[_0:=Ancestor v]> tail)
n: nat
hN: n < length (head :: tail)
<[n:=(256 - Z.of_nat (leafCount v))%Z]>
  (convertToArray (head :: tail)) =
convertToArray (<[n:=Ancestor v]> (head :: tail))
  destruct n as [| n].
head: Slot
tail: list Slot
v: Tree
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=(256 - Z.of_nat (leafCount v))%Z]>
      (convertToArray tail) =
    convertToArray (<[_0:=Ancestor v]> tail)
hN: 0 < length (head :: tail)
<[0:=(256 - Z.of_nat (leafCount v))%Z]>
  (convertToArray (head :: tail)) =
convertToArray (<[0:=Ancestor v]> (head :: tail))
head: Slot
tail: list Slot
v: Tree
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=(256 - Z.of_nat (leafCount v))%Z]>
      (convertToArray tail) =
    convertToArray (<[_0:=Ancestor v]> tail)
n: nat
hN: S n < length (head :: tail)
<[S n:=(256 - Z.of_nat (leafCount v))%Z]>
  (convertToArray (head :: tail)) =
convertToArray (<[S n:=Ancestor v]> (head :: tail))
  -
head: Slot
tail: list Slot
v: Tree
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=(256 - Z.of_nat (leafCount v))%Z]>
      (convertToArray tail) =
    convertToArray (<[_0:=Ancestor v]> tail)
hN: 0 < length (head :: tail)
<[0:=(256 - Z.of_nat (leafCount v))%Z]>
  (convertToArray (head :: tail)) =
convertToArray (<[0:=Ancestor v]> (head :: tail)) simpl.
head: Slot
tail: list Slot
v: Tree
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=(256 - Z.of_nat (leafCount v))%Z]>
      (convertToArray tail) =
    convertToArray (<[_0:=Ancestor v]> tail)
hN: 0 < length (head :: tail)
<[0:=(256 - Z.of_nat (leafCount v))%Z]>
  match head with
  | ReferTo _0 => Z.of_nat _0 :: convertToArray tail
  | Ancestor _0 =>
      (256 - Z.of_nat (leafCount _0))%Z
      :: convertToArray tail
  end =
(256 - Z.of_nat (leafCount v))%Z
:: convertToArray tail destruct head; simpl; reflexivity.
  -
head: Slot
tail: list Slot
v: Tree
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=(256 - Z.of_nat (leafCount v))%Z]>
      (convertToArray tail) =
    convertToArray (<[_0:=Ancestor v]> tail)
n: nat
hN: S n < length (head :: tail)
<[S n:=(256 - Z.of_nat (leafCount v))%Z]>
  (convertToArray (head :: tail)) =
convertToArray (<[S n:=Ancestor v]> (head :: tail)) simpl.
head: Slot
tail: list Slot
v: Tree
IH: ∀ _0 : nat,
  _0 < length tail
  → <[_0:=(256 - Z.of_nat (leafCount v))%Z]>
      (convertToArray tail) =
    convertToArray (<[_0:=Ancestor v]> tail)
n: nat
hN: S n < length (head :: tail)
<[S n:=(256 - Z.of_nat (leafCount v))%Z]>
  match head with
  | ReferTo _0 => Z.of_nat _0 :: convertToArray tail
  | Ancestor _0 =>
      (256 - Z.of_nat (leafCount _0))%Z
      :: convertToArray tail
  end =
match head with
| ReferTo _0 =>
    Z.of_nat _0
    :: convertToArray (<[n:=Ancestor v]> tail)
| Ancestor _0 =>
    (256 - Z.of_nat (leafCount _0))%Z
    :: convertToArray (<[n:=Ancestor v]> tail)
end destruct head; simpl; rewrite IH; try reflexivity; simpl in hN; lia.
Qed.

Lemma lengthConvert (dsu : list Slot) : length (convertToArray dsu) = length dsu.
dsu: list Slot
length (convertToArray dsu) = length dsu
Proof.
dsu: list Slot
length (convertToArray dsu) = length dsu
  induction dsu as [| [|] tail IH]; [easy | |]; simpl; now rewrite IH.
Qed.

Lemma nthUpperBoundConvertAux (dsu : list Slot) (h : length dsu < 256) (h1 : forall x y, nth x dsu (Ancestor Unit) = ReferTo y -> y < 256) (n : nat) : Z.lt (nth n (convertToArray dsu) (0%Z)) (256%Z).
dsu: list Slot
h: length dsu < 256
h1: ∀ _0 _1 : nat,
  nth _0 dsu (Ancestor Unit) = ReferTo _1
  → _1 < 256
n: nat
(nth n (convertToArray dsu) 0 < 256)%Z
Proof.
dsu: list Slot
h: length dsu < 256
h1: ∀ _0 _1 : nat,
  nth _0 dsu (Ancestor Unit) = ReferTo _1
  → _1 < 256
n: nat
(nth n (convertToArray dsu) 0 < 256)%Z
  revert n.
dsu: list Slot
h: length dsu < 256
h1: ∀ _0 _1 : nat,
  nth _0 dsu (Ancestor Unit) = ReferTo _1
  → _1 < 256
∀ _0 : nat, (nth _0 (convertToArray dsu) 0 < 256)%Z induction dsu as [| [a | a] tail IH]; intro n; [destruct n; easy | |]; destruct n as [| n].
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
(nth 0 (convertToArray (ReferTo a :: tail)) 0 < 256)%Z
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
(nth (S n) (convertToArray (ReferTo a :: tail)) 0 <
 256)%Z
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
(nth 0 (convertToArray (Ancestor a :: tail)) 0 < 256)%Z
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
(nth (S n) (convertToArray (Ancestor a :: tail)) 0 <
 256)%Z
  -
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
(nth 0 (convertToArray (ReferTo a :: tail)) 0 < 256)%Z simpl.
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
(Z.of_nat a < 256)%Z pose proof h1 0 a ltac:(easy).
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
_0: a < 256
(Z.of_nat a < 256)%Z lia.
  -
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
(nth (S n) (convertToArray (ReferTo a :: tail)) 0 <
 256)%Z simpl.
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
(nth n (convertToArray tail) 0 < 256)%Z apply IH.
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
length tail < 256
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
∀ _0 _1 : nat,
  nth _0 tail (Ancestor Unit) = ReferTo _1 → _1 < 256 {
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
length tail < 256 simpl in h; lia. }
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
∀ _0 _1 : nat,
  nth _0 tail (Ancestor Unit) = ReferTo _1 → _1 < 256 intros s t.
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n, s, t: nat
nth s tail (Ancestor Unit) = ReferTo t → t < 256 pose proof h1 (S s) t as step.
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n, s, t: nat
step: nth (S s) (ReferTo a :: tail) (Ancestor Unit) =
ReferTo t → t < 256
nth s tail (Ancestor Unit) = ReferTo t → t < 256 simpl in step.
a: nat
tail: list Slot
h: length (ReferTo a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (ReferTo a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n, s, t: nat
step: nth s tail (Ancestor Unit) = ReferTo t
→ t < 256
nth s tail (Ancestor Unit) = ReferTo t → t < 256 exact step.
  -
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
(nth 0 (convertToArray (Ancestor a :: tail)) 0 < 256)%Z simpl.
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
(256 - Z.of_nat (leafCount a) < 256)%Z pose proof oneLeqLeafCount a.
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
_0: 1 ≤ leafCount a
(256 - Z.of_nat (leafCount a) < 256)%Z lia.
  -
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
(nth (S n) (convertToArray (Ancestor a :: tail)) 0 <
 256)%Z simpl.
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
(nth n (convertToArray tail) 0 < 256)%Z apply IH.
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
length tail < 256
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
∀ _0 _1 : nat,
  nth _0 tail (Ancestor Unit) = ReferTo _1 → _1 < 256 {
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
length tail < 256 simpl in h; lia. }
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n: nat
∀ _0 _1 : nat,
  nth _0 tail (Ancestor Unit) = ReferTo _1 → _1 < 256 intros s t.
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n, s, t: nat
nth s tail (Ancestor Unit) = ReferTo t → t < 256 pose proof h1 (S s) t as step.
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n, s, t: nat
step: nth (S s) (Ancestor a :: tail) (Ancestor Unit) =
ReferTo t → t < 256
nth s tail (Ancestor Unit) = ReferTo t → t < 256 simpl in step.
a: Tree
tail: list Slot
h: length (Ancestor a :: tail) < 256
h1: ∀ _0 _1 : nat,
  nth _0 (Ancestor a :: tail) (Ancestor Unit) =
  ReferTo _1 → _1 < 256
IH: length tail < 256
→ (∀ _0 _1 : nat,
     nth _0 tail (Ancestor Unit) = ReferTo _1
     → _1 < 256)
  → ∀ _0 : nat,
      (nth _0 (convertToArray tail) 0 < 256)%Z
n, s, t: nat
step: nth s tail (Ancestor Unit) = ReferTo t
→ t < 256
nth s tail (Ancestor Unit) = ReferTo t → t < 256 exact step.
Qed.

Lemma nthUpperBoundConvert (dsu : list Slot) (h : length dsu < 256) (h1 : noIllegalIndices dsu) (n : nat) : Z.lt (nth n (convertToArray dsu) (0%Z)) (256%Z).
dsu: list Slot
h: length dsu < 256
h1: noIllegalIndices dsu
n: nat
(nth n (convertToArray dsu) 0 < 256)%Z
Proof.
dsu: list Slot
h: length dsu < 256
h1: noIllegalIndices dsu
n: nat
(nth n (convertToArray dsu) 0 < 256)%Z
  apply nthUpperBoundConvertAux; try assumption.
dsu: list Slot
h: length dsu < 256
h1: noIllegalIndices dsu
n: nat
∀ _0 _1 : nat,
  nth _0 dsu (Ancestor Unit) = ReferTo _1 → _1 < 256
  intros a b c.
dsu: list Slot
h: length dsu < 256
h1: noIllegalIndices dsu
n, a, b: nat
c: nth a dsu (Ancestor Unit) = ReferTo b
b < 256 pose proof h1 a b c.
dsu: list Slot
h: length dsu < 256
h1: noIllegalIndices dsu
n, a, b: nat
c: nth a dsu (Ancestor Unit) = ReferTo b
_0: b < length dsu
b < 256 lia.
Qed.

Definition dsuScore (dsu : list Slot) := Z.of_nat (list_sum (map (fun x => match x with | ReferTo _ => 0 | Ancestor x => score x end) dsu)).

Definition dsuLeafCount (dsu : list Slot) := Z.of_nat (list_sum (map (fun x => match x with | ReferTo _ => 0 | Ancestor x => leafCount x end) dsu)).

Lemma sumTwoAncestors' (dsu : list Slot) (a b : nat) (hAB : a < b) (hB : b < length dsu) u (hA1 : nth a dsu (Ancestor Unit) = Ancestor u) v (hB1 : nth b dsu (Ancestor Unit) = Ancestor v) : leafCount u + leafCount v <= Z.to_nat (dsuLeafCount dsu).
dsu: list Slot
a, b: nat
hAB: a < b
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
leafCount u + leafCount v
≤ Z.to_nat (dsuLeafCount dsu)
Proof.
dsu: list Slot
a, b: nat
hAB: a < b
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
leafCount u + leafCount v
≤ Z.to_nat (dsuLeafCount dsu)
  pose proof ListDecomposition.listDecomposition dsu a b hAB hB (Ancestor Unit) as step.
dsu: list Slot
a, b: nat
hAB: a < b
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
step: dsu =
take a dsu ++
[nth a dsu (Ancestor Unit)] ++
drop (S a) (take b dsu) ++
[nth b dsu (Ancestor Unit)] ++ drop (b + 1) dsu
leafCount u + leafCount v
≤ Z.to_nat (dsuLeafCount dsu) rewrite step.
dsu: list Slot
a, b: nat
hAB: a < b
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
step: dsu =
take a dsu ++
[nth a dsu (Ancestor Unit)] ++
drop (S a) (take b dsu) ++
[nth b dsu (Ancestor Unit)] ++ drop (b + 1) dsu
leafCount u + leafCount v
≤ Z.to_nat
    (dsuLeafCount
       (take a dsu ++
        [nth a dsu (Ancestor Unit)] ++
        drop (S a) (take b dsu) ++
        [nth b dsu (Ancestor Unit)] ++
        drop (b + 1) dsu)) unfold dsuLeafCount.
dsu: list Slot
a, b: nat
hAB: a < b
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
step: dsu =
take a dsu ++
[nth a dsu (Ancestor Unit)] ++
drop (S a) (take b dsu) ++
[nth b dsu (Ancestor Unit)] ++ drop (b + 1) dsu
leafCount u + leafCount v
≤ Z.to_nat
    (Z.of_nat
       (list_sum
          (map
             (λ _0 : Slot,
                match _0 with
                | ReferTo _ => 0
                | Ancestor _1 => leafCount _1
                end)
             (take a dsu ++
              [nth a dsu (Ancestor Unit)] ++
              drop (S a) (take b dsu) ++
              [nth b dsu (Ancestor Unit)] ++
              drop (b + 1) dsu)))) rewrite Nat2Z.id, !map_app, hA1, hB1, !list_sum_app.
dsu: list Slot
a, b: nat
hAB: a < b
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
step: dsu =
take a dsu ++
[nth a dsu (Ancestor Unit)] ++
drop (S a) (take b dsu) ++
[nth b dsu (Ancestor Unit)] ++ drop (b + 1) dsu
leafCount u + leafCount v
≤ list_sum
    (map
       (λ _0 : Slot,
          match _0 with
          | ReferTo _ => 0
          | Ancestor _1 => leafCount _1
          end) (take a dsu)) +
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) [Ancestor u]) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S a) (take b dsu))) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) [Ancestor v]) +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (b + 1) dsu))))) simpl.
dsu: list Slot
a, b: nat
hAB: a < b
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
step: dsu =
take a dsu ++
[nth a dsu (Ancestor Unit)] ++
drop (S a) (take b dsu) ++
[nth b dsu (Ancestor Unit)] ++ drop (b + 1) dsu
leafCount u + leafCount v
≤ list_sum
    (map
       (λ _0 : Slot,
          match _0 with
          | ReferTo _ => 0
          | Ancestor _1 => leafCount _1
          end) (take a dsu)) +
  (leafCount u + 0 +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S a) (take b dsu))) +
    (leafCount v + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (b + 1) dsu))))) lia.
Qed.

Lemma sumTwoAncestors (dsu : list Slot) (a b : nat) (hAB : a <> b) (hA : a < length dsu) (hB : b < length dsu) u (hA1 : nth a dsu (Ancestor Unit) = Ancestor u) v (hB1 : nth b dsu (Ancestor Unit) = Ancestor v) : leafCount u + leafCount v <= Z.to_nat (dsuLeafCount dsu).
dsu: list Slot
a, b: nat
hAB: a ≠ b
hA: a < length dsu
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
leafCount u + leafCount v
≤ Z.to_nat (dsuLeafCount dsu)
Proof.
dsu: list Slot
a, b: nat
hAB: a ≠ b
hA: a < length dsu
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
leafCount u + leafCount v
≤ Z.to_nat (dsuLeafCount dsu)
  destruct (ltac:(lia) : a < b \/ b < a) as [hs | hs]; [| rewrite Nat.add_comm].
dsu: list Slot
a, b: nat
hAB: a ≠ b
hA: a < length dsu
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
hs: a < b
leafCount u + leafCount v
≤ Z.to_nat (dsuLeafCount dsu)
dsu: list Slot
a, b: nat
hAB: a ≠ b
hA: a < length dsu
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
hs: b < a
leafCount v + leafCount u
≤ Z.to_nat (dsuLeafCount dsu)
  -
dsu: list Slot
a, b: nat
hAB: a ≠ b
hA: a < length dsu
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
hs: a < b
leafCount u + leafCount v
≤ Z.to_nat (dsuLeafCount dsu) apply (sumTwoAncestors' _ a b); assumption.
  -
dsu: list Slot
a, b: nat
hAB: a ≠ b
hA: a < length dsu
hB: b < length dsu
u: Tree
hA1: nth a dsu (Ancestor Unit) = Ancestor u
v: Tree
hB1: nth b dsu (Ancestor Unit) = Ancestor v
hs: b < a
leafCount v + leafCount u
≤ Z.to_nat (dsuLeafCount dsu) apply (sumTwoAncestors' _ b a); assumption.
Qed.

Lemma nthLowerBoundConvertAuxStep (dsu : list Slot) (h : length dsu < 256) (h1 : Z.to_nat (dsuLeafCount dsu) < 128) (n : nat) (hn : n < length dsu) x (h2 : nth n dsu (Ancestor Unit) = Ancestor x) : leafCount x < 128.
dsu: list Slot
h: length dsu < 256
h1: Z.to_nat (dsuLeafCount dsu) < 128
n: nat
hn: n < length dsu
x: Tree
h2: nth n dsu (Ancestor Unit) = Ancestor x
leafCount x < 128
Proof.
dsu: list Slot
h: length dsu < 256
h1: Z.to_nat (dsuLeafCount dsu) < 128
n: nat
hn: n < length dsu
x: Tree
h2: nth n dsu (Ancestor Unit) = Ancestor x
leafCount x < 128
  revert n hn x h2.
dsu: list Slot
h: length dsu < 256
h1: Z.to_nat (dsuLeafCount dsu) < 128
∀ _0 : nat,
  _0 < length dsu
  → ∀ _1 : Tree,
      nth _0 dsu (Ancestor Unit) = Ancestor _1
      → leafCount _1 < 128 induction dsu as [| head tail IH].
h: length [] < 256
h1: Z.to_nat (dsuLeafCount []) < 128
∀ _0 : nat,
  _0 < length []
  → ∀ _1 : Tree,
      nth _0 [] (Ancestor Unit) = Ancestor _1
      → leafCount _1 < 128
head: Slot
tail: list Slot
h: length (head :: tail) < 256
h1: Z.to_nat (dsuLeafCount (head :: tail)) < 128
IH: length tail < 256
→ Z.to_nat (dsuLeafCount tail) < 128
  → ∀ _0 : nat,
      _0 < length tail
      → ∀ _1 : Tree,
          nth _0 tail (Ancestor Unit) =
          Ancestor _1 → leafCount _1 < 128
∀ _0 : nat,
  _0 < length (head :: tail)
  → ∀ _1 : Tree,
      nth _0 (head :: tail) (Ancestor Unit) =
      Ancestor _1 → leafCount _1 < 128
  {
h: length [] < 256
h1: Z.to_nat (dsuLeafCount []) < 128
∀ _0 : nat,
  _0 < length []
  → ∀ _1 : Tree,
      nth _0 [] (Ancestor Unit) = Ancestor _1
      → leafCount _1 < 128 simpl.
h: length [] < 256
h1: Z.to_nat (dsuLeafCount []) < 128
∀ _0 : nat,
  _0 < 0
  → ∀ _1 : Tree,
      match _0 with
      | 0 | _ => Ancestor Unit
      end = Ancestor _1 → 
      leafCount _1 < 128 intro.
h: length [] < 256
h1: Z.to_nat (dsuLeafCount []) < 128
_0: nat
_0 < 0
→ ∀ _1 : Tree,
    match _0 with
    | 0 | _ => Ancestor Unit
    end = Ancestor _1 → leafCount _1 < 128 lia. }
head: Slot
tail: list Slot
h: length (head :: tail) < 256
h1: Z.to_nat (dsuLeafCount (head :: tail)) < 128
IH: length tail < 256
→ Z.to_nat (dsuLeafCount tail) < 128
  → ∀ _0 : nat,
      _0 < length tail
      → ∀ _1 : Tree,
          nth _0 tail (Ancestor Unit) =
          Ancestor _1 → leafCount _1 < 128
∀ _0 : nat,
  _0 < length (head :: tail)
  → ∀ _1 : Tree,
      nth _0 (head :: tail) (Ancestor Unit) =
      Ancestor _1 → leafCount _1 < 128
  intros n h2 h3 h4.
head: Slot
tail: list Slot
h: length (head :: tail) < 256
h1: Z.to_nat (dsuLeafCount (head :: tail)) < 128
IH: length tail < 256
→ Z.to_nat (dsuLeafCount tail) < 128
  → ∀ _0 : nat,
      _0 < length tail
      → ∀ _1 : Tree,
          nth _0 tail (Ancestor Unit) =
          Ancestor _1 → leafCount _1 < 128
n: nat
h2: n < length (head :: tail)
h3: Tree
h4: nth n (head :: tail) (Ancestor Unit) =
Ancestor h3
leafCount h3 < 128
  destruct n as [| n].
head: Slot
tail: list Slot
h: length (head :: tail) < 256
h1: Z.to_nat (dsuLeafCount (head :: tail)) < 128
IH: length tail < 256
→ Z.to_nat (dsuLeafCount tail) < 128
  → ∀ _0 : nat,
      _0 < length tail
      → ∀ _1 : Tree,
          nth _0 tail (Ancestor Unit) =
          Ancestor _1 → leafCount _1 < 128
h2: 0 < length (head :: tail)
h3: Tree
h4: nth 0 (head :: tail) (Ancestor Unit) =
Ancestor h3
leafCount h3 < 128
head: Slot
tail: list Slot
h: length (head :: tail) < 256
h1: Z.to_nat (dsuLeafCount (head :: tail)) < 128
IH: length tail < 256
→ Z.to_nat (dsuLeafCount tail) < 128
  → ∀ _0 : nat,
      _0 < length tail
      → ∀ _1 : Tree,
          nth _0 tail (Ancestor Unit) =
          Ancestor _1 → leafCount _1 < 128
n: nat
h2: S n < length (head :: tail)
h3: Tree
h4: nth (S n) (head :: tail) (Ancestor Unit) =
Ancestor h3
leafCount h3 < 128
  -
head: Slot
tail: list Slot
h: length (head :: tail) < 256
h1: Z.to_nat (dsuLeafCount (head :: tail)) < 128
IH: length tail < 256
→ Z.to_nat (dsuLeafCount tail) < 128
  → ∀ _0 : nat,
      _0 < length tail
      → ∀ _1 : Tree,
          nth _0 tail (Ancestor Unit) =
          Ancestor _1 → leafCount _1 < 128
h2: 0 < length (head :: tail)
h3: Tree
h4: nth 0 (head :: tail) (Ancestor Unit) =
Ancestor h3
leafCount h3 < 128 simpl in h4.
head: Slot
tail: list Slot
h: length (head :: tail) < 256
h1: Z.to_nat (dsuLeafCount (head :: tail)) < 128
IH: length tail < 256
→ Z.to_nat (dsuLeafCount tail) < 128
  → ∀ _0 : nat,
      _0 < length tail
      → ∀ _1 : Tree,
          nth _0 tail (Ancestor Unit) =
          Ancestor _1 → leafCount _1 < 128
h2: 0 < length (head :: tail)
h3: Tree
h4: head = Ancestor h3
leafCount h3 < 128 unfold dsuLeafCount in h1.
head: Slot
tail: list Slot
h: length (head :: tail) < 256
h1: Z.to_nat
  (Z.of_nat
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (head :: tail)))) < 128
IH: length tail < 256
→ Z.to_nat (dsuLeafCount tail) < 128
  → ∀ _0 : nat,
      _0 < length tail
      → ∀ _1 : Tree,
          nth _0 tail (Ancestor Unit) =
          Ancestor _1 → leafCount _1 < 128
h2: 0 < length (head :: tail)
h3: Tree
h4: head = Ancestor h3
leafCount h3 < 128 rewrite -> Nat2Z.id, map_cons, (ltac:(simpl; reflexivity) : list_sum (_ :: _) = _ + list_sum _) in h1.
head: Slot
tail: list Slot
h: length (head :: tail) < 256
h1: match head with
| ReferTo _ => 0
| Ancestor _0 => leafCount _0
end +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) < 128
IH: length tail < 256
→ Z.to_nat (dsuLeafCount tail) < 128
  → ∀ _0 : nat,
      _0 < length tail
      → ∀ _1 : Tree,
          nth _0 tail (Ancestor Unit) =
          Ancestor _1 → leafCount _1 < 128
h2: 0 < length (head :: tail)
h3: Tree
h4: head = Ancestor h3
leafCount h3 < 128 rewrite h4 in h1.
head: Slot
tail: list Slot
h: length (head :: tail) < 256
h3: Tree
h1: leafCount h3 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) < 128
IH: length tail < 256
→ Z.to_nat (dsuLeafCount tail) < 128
  → ∀ _0 : nat,
      _0 < length tail
      → ∀ _1 : Tree,
          nth _0 tail (Ancestor Unit) =
          Ancestor _1 → leafCount _1 < 128
h2: 0 < length (head :: tail)
h4: head = Ancestor h3
leafCount h3 < 128 lia.
  -
head: Slot
tail: list Slot
h: length (head :: tail) < 256
h1: Z.to_nat (dsuLeafCount (head :: tail)) < 128
IH: length tail < 256
→ Z.to_nat (dsuLeafCount tail) < 128
  → ∀ _0 : nat,
      _0 < length tail
      → ∀ _1 : Tree,
          nth _0 tail (Ancestor Unit) =
          Ancestor _1 → leafCount _1 < 128
n: nat
h2: S n < length (head :: tail)
h3: Tree
h4: nth (S n) (head :: tail) (Ancestor Unit) =
Ancestor h3
leafCount h3 < 128 exact (IH ltac:(simpl in h; lia) ltac:(unfold dsuLeafCount in h1; rewrite Nat2Z.id in h1; rewrite -> map_cons, (ltac:(simpl; reflexivity) : list_sum (_ :: _) = _ + list_sum _) in h1; unfold dsuLeafCount; rewrite Nat2Z.id; lia) n ltac:(simpl in h2; lia) h3 ltac:(simpl in h4; exact h4)).
Qed.

Lemma nthLowerBoundConvertAux (dsu : list Slot) (h : length dsu < 256) (h1 : Z.to_nat (dsuLeafCount dsu) < 128) (n : nat) : Z.le 0%Z (nth n (convertToArray dsu) (0%Z)).
dsu: list Slot
h: length dsu < 256
h1: Z.to_nat (dsuLeafCount dsu) < 128
n: nat
(0 ≤ nth n (convertToArray dsu) 0)%Z
Proof.
dsu: list Slot
h: length dsu < 256
h1: Z.to_nat (dsuLeafCount dsu) < 128
n: nat
(0 ≤ nth n (convertToArray dsu) 0)%Z
  revert n h1.
dsu: list Slot
h: length dsu < 256
∀ _0 : nat,
  Z.to_nat (dsuLeafCount dsu) < 128
  → (0 ≤ nth _0 (convertToArray dsu) 0)%Z induction dsu as [| head tail IH].
h: length [] < 256
∀ _0 : nat,
  Z.to_nat (dsuLeafCount []) < 128
  → (0 ≤ nth _0 (convertToArray []) 0)%Z
head: Slot
tail: list Slot
h: length (head :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
∀ _0 : nat,
  Z.to_nat (dsuLeafCount (head :: tail)) < 128
  → (0 ≤ nth _0 (convertToArray (head :: tail)) 0)%Z
  {
h: length [] < 256
∀ _0 : nat,
  Z.to_nat (dsuLeafCount []) < 128
  → (0 ≤ nth _0 (convertToArray []) 0)%Z cbv.
h: length [] < 256
∀ _0 : nat,
  1 ≤ 128
  → match match _0 with
          | 0 | _ => 0%Z
          end with
    | 0%Z => Eq
    | Z.pos _ => Lt
    | Z.neg _ => Gt
    end = Gt → False intros a b.
h: length [] < 256
a: nat
b: 1 ≤ 128
match match a with
      | 0 | _ => 0%Z
      end with
| 0%Z => Eq
| Z.pos _ => Lt
| Z.neg _ => Gt
end = Gt → False destruct a; easy. }
head: Slot
tail: list Slot
h: length (head :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
∀ _0 : nat,
  Z.to_nat (dsuLeafCount (head :: tail)) < 128
  → (0 ≤ nth _0 (convertToArray (head :: tail)) 0)%Z
  intros [| a] b.
head: Slot
tail: list Slot
h: length (head :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
b: Z.to_nat (dsuLeafCount (head :: tail)) < 128
(0 ≤ nth 0 (convertToArray (head :: tail)) 0)%Z
head: Slot
tail: list Slot
h: length (head :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
a: nat
b: Z.to_nat (dsuLeafCount (head :: tail)) < 128
(0 ≤ nth (S a) (convertToArray (head :: tail)) 0)%Z
  -
head: Slot
tail: list Slot
h: length (head :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
b: Z.to_nat (dsuLeafCount (head :: tail)) < 128
(0 ≤ nth 0 (convertToArray (head :: tail)) 0)%Z simpl.
head: Slot
tail: list Slot
h: length (head :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
b: Z.to_nat (dsuLeafCount (head :: tail)) < 128
(0
 ≤ nth 0
     match head with
     | ReferTo _0 =>
         Z.of_nat _0 :: convertToArray tail
     | Ancestor _0 =>
         256 - Z.of_nat (leafCount _0)
         :: convertToArray tail
     end 0)%Z destruct head as [x | x].
x: nat
tail: list Slot
h: length (ReferTo x :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
b: Z.to_nat (dsuLeafCount (ReferTo x :: tail)) < 128
(0 ≤ nth 0 (Z.of_nat x :: convertToArray tail) 0)%Z
x: Tree
tail: list Slot
h: length (Ancestor x :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
b: Z.to_nat (dsuLeafCount (Ancestor x :: tail)) < 128
(0
 ≤ nth 0
     (256 - Z.of_nat (leafCount x)
      :: convertToArray tail) 0)%Z {
x: nat
tail: list Slot
h: length (ReferTo x :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
b: Z.to_nat (dsuLeafCount (ReferTo x :: tail)) < 128
(0 ≤ nth 0 (Z.of_nat x :: convertToArray tail) 0)%Z simpl.
x: nat
tail: list Slot
h: length (ReferTo x :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
b: Z.to_nat (dsuLeafCount (ReferTo x :: tail)) < 128
(0 ≤ Z.of_nat x)%Z lia. }
x: Tree
tail: list Slot
h: length (Ancestor x :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
b: Z.to_nat (dsuLeafCount (Ancestor x :: tail)) < 128
(0
 ≤ nth 0
     (256 - Z.of_nat (leafCount x)
      :: convertToArray tail) 0)%Z
    simpl.
x: Tree
tail: list Slot
h: length (Ancestor x :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
b: Z.to_nat (dsuLeafCount (Ancestor x :: tail)) < 128
(0 ≤ 256 - Z.of_nat (leafCount x))%Z unfold dsuLeafCount in b.
x: Tree
tail: list Slot
h: length (Ancestor x :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
b: Z.to_nat
  (Z.of_nat
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (Ancestor x :: tail)))) < 128
(0 ≤ 256 - Z.of_nat (leafCount x))%Z rewrite -> Nat2Z.id, map_cons, (ltac:(simpl; reflexivity) : list_sum (_ :: _) = _ + list_sum _) in b.
x: Tree
tail: list Slot
h: length (Ancestor x :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
b: leafCount x +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) < 128
(0 ≤ 256 - Z.of_nat (leafCount x))%Z lia.
  -
head: Slot
tail: list Slot
h: length (head :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
a: nat
b: Z.to_nat (dsuLeafCount (head :: tail)) < 128
(0 ≤ nth (S a) (convertToArray (head :: tail)) 0)%Z rewrite (ltac:(simpl; destruct head; easy) : nth (S a) (convertToArray (head :: tail)) (0%Z) = nth a (convertToArray tail) (0%Z)).
head: Slot
tail: list Slot
h: length (head :: tail) < 256
IH: length tail < 256
→ ∀ _0 : nat,
    Z.to_nat (dsuLeafCount tail) < 128
    → (0 ≤ nth _0 (convertToArray tail) 0)%Z
a: nat
b: Z.to_nat (dsuLeafCount (head :: tail)) < 128
(0 ≤ nth a (convertToArray tail) 0)%Z
    exact (IH ltac:(simpl in h; lia) a ltac:(unfold dsuLeafCount in b; rewrite -> Nat2Z.id, map_cons, (ltac:(simpl; reflexivity) : list_sum (_ :: _) = _ + list_sum _) in b; unfold dsuLeafCount; rewrite Nat2Z.id; lia)).
Qed.

Lemma nthLowerBoundConvert (dsu : list Slot) (h : length dsu < 128) (h1 : Z.to_nat (dsuLeafCount dsu) = length dsu) (n : nat) : Z.le 0%Z (nth n (convertToArray dsu) (0%Z)).
dsu: list Slot
h: length dsu < 128
h1: Z.to_nat (dsuLeafCount dsu) = length dsu
n: nat
(0 ≤ nth n (convertToArray dsu) 0)%Z
Proof.
dsu: list Slot
h: length dsu < 128
h1: Z.to_nat (dsuLeafCount dsu) = length dsu
n: nat
(0 ≤ nth n (convertToArray dsu) 0)%Z apply nthLowerBoundConvertAux; try (assumption || lia). Qed.

Fixpoint ancestor (dsu : list Slot) (fuel : nat) (index : nat) :=
  match fuel with
  | O => index
  | S fuel => match nth index dsu (Ancestor Unit) with
    | ReferTo x => ancestor dsu fuel x
    | Ancestor _ => index
    end
  end.

Fixpoint ancestorChain (dsu : list Slot) (fuel : nat) (index : nat) :=
  match fuel with
  | O => [index]
  | S fuel => match nth index dsu (Ancestor Unit) with
    | ReferTo x => index :: ancestorChain dsu fuel x
    | Ancestor _ => index :: nil
    end
  end.

Lemma firstAncestorChain dsu fuel index : nth 0 (ancestorChain dsu fuel index) 0 = index.
dsu: list Slot
fuel, index: nat
nth 0 (ancestorChain dsu fuel index) 0 = index
Proof.
dsu: list Slot
fuel, index: nat
nth 0 (ancestorChain dsu fuel index) 0 = index
  destruct fuel as [| fuel].
dsu: list Slot
index: nat
nth 0 (ancestorChain dsu 0 index) 0 = index
dsu: list Slot
fuel, index: nat
nth 0 (ancestorChain dsu (S fuel) index) 0 = index {
dsu: list Slot
index: nat
nth 0 (ancestorChain dsu 0 index) 0 = index easy. }
dsu: list Slot
fuel, index: nat
nth 0 (ancestorChain dsu (S fuel) index) 0 = index
  simpl.
dsu: list Slot
fuel, index: nat
nth 0
  match nth index dsu (Ancestor Unit) with
  | ReferTo _0 => index :: ancestorChain dsu fuel _0
  | Ancestor _ => [index]
  end 0 = index destruct (nth index _ _); easy.
Qed.

Lemma zeroLtLengthAncestorChain dsu fuel index : 0 < length (ancestorChain dsu fuel index).
dsu: list Slot
fuel, index: nat
0 < length (ancestorChain dsu fuel index)
Proof.
dsu: list Slot
fuel, index: nat
0 < length (ancestorChain dsu fuel index)
  destruct fuel as [| fuel].
dsu: list Slot
index: nat
0 < length (ancestorChain dsu 0 index)
dsu: list Slot
fuel, index: nat
0 < length (ancestorChain dsu (S fuel) index) {
dsu: list Slot
index: nat
0 < length (ancestorChain dsu 0 index) simpl.
dsu: list Slot
index: nat
0 < 1 lia. }
dsu: list Slot
fuel, index: nat
0 < length (ancestorChain dsu (S fuel) index)
  simpl.
dsu: list Slot
fuel, index: nat
0 <
length
  match nth index dsu (Ancestor Unit) with
  | ReferTo _0 => index :: ancestorChain dsu fuel _0
  | Ancestor _ => [index]
  end destruct (nth index _ _); cbv; lia.
Qed.


Lemma ancestorEqLastAncestorChain' dsu fuel index : last (ancestorChain dsu fuel index) = Some (ancestor dsu fuel index).
dsu: list Slot
fuel, index: nat
last (ancestorChain dsu fuel index) =
Some (ancestor dsu fuel index)
Proof.
dsu: list Slot
fuel, index: nat
last (ancestorChain dsu fuel index) =
Some (ancestor dsu fuel index)
  induction fuel as [| fuel IH] in index |- *.
dsu: list Slot
index: nat
last (ancestorChain dsu 0 index) =
Some (ancestor dsu 0 index)
dsu: list Slot
fuel, index: nat
IH: ∀ _0 : nat,
  last (ancestorChain dsu fuel _0) =
  Some (ancestor dsu fuel _0)
last (ancestorChain dsu (S fuel) index) =
Some (ancestor dsu (S fuel) index) {
dsu: list Slot
index: nat
last (ancestorChain dsu 0 index) =
Some (ancestor dsu 0 index) easy. }
dsu: list Slot
fuel, index: nat
IH: ∀ _0 : nat,
  last (ancestorChain dsu fuel _0) =
  Some (ancestor dsu fuel _0)
last (ancestorChain dsu (S fuel) index) =
Some (ancestor dsu (S fuel) index)
  simpl.
dsu: list Slot
fuel, index: nat
IH: ∀ _0 : nat,
  last (ancestorChain dsu fuel _0) =
  Some (ancestor dsu fuel _0)
last
  match nth index dsu (Ancestor Unit) with
  | ReferTo _0 => index :: ancestorChain dsu fuel _0
  | Ancestor _ => [index]
  end =
Some
  match nth index dsu (Ancestor Unit) with
  | ReferTo _0 => ancestor dsu fuel _0
  | Ancestor _ => index
  end remember (nth index dsu (Ancestor Unit)) as x eqn:hX.
dsu: list Slot
fuel, index: nat
IH: ∀ _0 : nat,
  last (ancestorChain dsu fuel _0) =
  Some (ancestor dsu fuel _0)
x: Slot
hX: x = nth index dsu (Ancestor Unit)
last
  match x with
  | ReferTo _0 => index :: ancestorChain dsu fuel _0
  | Ancestor _ => [index]
  end =
Some
  match x with
  | ReferTo _0 => ancestor dsu fuel _0
  | Ancestor _ => index
  end destruct x as [x | x]; symmetry in hX.
dsu: list Slot
fuel, index: nat
IH: ∀ _0 : nat,
  last (ancestorChain dsu fuel _0) =
  Some (ancestor dsu fuel _0)
x: nat
hX: nth index dsu (Ancestor Unit) = ReferTo x
last (index :: ancestorChain dsu fuel x) =
Some (ancestor dsu fuel x)
dsu: list Slot
fuel, index: nat
IH: ∀ _0 : nat,
  last (ancestorChain dsu fuel _0) =
  Some (ancestor dsu fuel _0)
x: Tree
hX: nth index dsu (Ancestor Unit) = Ancestor x
last [index] = Some index
  -
dsu: list Slot
fuel, index: nat
IH: ∀ _0 : nat,
  last (ancestorChain dsu fuel _0) =
  Some (ancestor dsu fuel _0)
x: nat
hX: nth index dsu (Ancestor Unit) = ReferTo x
last (index :: ancestorChain dsu fuel x) =
Some (ancestor dsu fuel x) rewrite last_cons.
dsu: list Slot
fuel, index: nat
IH: ∀ _0 : nat,
  last (ancestorChain dsu fuel _0) =
  Some (ancestor dsu fuel _0)
x: nat
hX: nth index dsu (Ancestor Unit) = ReferTo x
match last (ancestorChain dsu fuel x) with
| Some _0 => Some _0
| None => Some index
end = Some (ancestor dsu fuel x) pose proof IH x as h.
dsu: list Slot
fuel, index: nat
IH: ∀ _0 : nat,
  last (ancestorChain dsu fuel _0) =
  Some (ancestor dsu fuel _0)
x: nat
hX: nth index dsu (Ancestor Unit) = ReferTo x
h: last (ancestorChain dsu fuel x) =
Some (ancestor dsu fuel x)
match last (ancestorChain dsu fuel x) with
| Some _0 => Some _0
| None => Some index
end = Some (ancestor dsu fuel x) now rewrite h.
  -
dsu: list Slot
fuel, index: nat
IH: ∀ _0 : nat,
  last (ancestorChain dsu fuel _0) =
  Some (ancestor dsu fuel _0)
x: Tree
hX: nth index dsu (Ancestor Unit) = Ancestor x
last [index] = Some index easy.
Qed.

Lemma defaultLast {A : Type} (l : list A) (d : A) : default d (last l) = nth (length l - 1) l d.
A: Type
l: list A
d: A
default d (last l) = nth (length l - 1) l d
Proof.
A: Type
l: list A
d: A
default d (last l) = nth (length l - 1) l d
  induction l as [| head tail IH].
A: Type
d: A
default d (last []) = nth (length [] - 1) [] d
A: Type
head: A
tail: list A
d: A
IH: default d (last tail) =
nth (length tail - 1) tail d
default d (last (head :: tail)) =
nth (length (head :: tail) - 1) (head :: tail) d {
A: Type
d: A
default d (last []) = nth (length [] - 1) [] d easy. }
A: Type
head: A
tail: list A
d: A
IH: default d (last tail) =
nth (length tail - 1) tail d
default d (last (head :: tail)) =
nth (length (head :: tail) - 1) (head :: tail) d
  simpl.
A: Type
head: A
tail: list A
d: A
IH: default d (last tail) =
nth (length tail - 1) tail d
default d (last (head :: tail)) =
match length tail - 0 with
| 0 => head
| S _0 => nth _0 tail d
end rewrite last_cons.
A: Type
head: A
tail: list A
d: A
IH: default d (last tail) =
nth (length tail - 1) tail d
default d
  match last tail with
  | Some _0 => Some _0
  | None => Some head
  end =
match length tail - 0 with
| 0 => head
| S _0 => nth _0 tail d
end rewrite Nat.sub_0_r.
A: Type
head: A
tail: list A
d: A
IH: default d (last tail) =
nth (length tail - 1) tail d
default d
  match last tail with
  | Some _0 => Some _0
  | None => Some head
  end =
match length tail with
| 0 => head
| S _0 => nth _0 tail d
end
  destruct tail as [| head' tail'].
A: Type
head, d: A
IH: default d (last []) = nth (length [] - 1) [] d
default d
  match last [] with
  | Some _0 => Some _0
  | None => Some head
  end =
match length [] with
| 0 => head
| S _0 => nth _0 [] d
end
A: Type
head, head': A
tail': list A
d: A
IH: default d (last (head' :: tail')) =
nth (length (head' :: tail') - 1)
  (head' :: tail') d
default d
  match last (head' :: tail') with
  | Some _0 => Some _0
  | None => Some head
  end =
match length (head' :: tail') with
| 0 => head
| S _0 => nth _0 (head' :: tail') d
end
  {
A: Type
head, d: A
IH: default d (last []) = nth (length [] - 1) [] d
default d
  match last [] with
  | Some _0 => Some _0
  | None => Some head
  end =
match length [] with
| 0 => head
| S _0 => nth _0 [] d
end easy. }
A: Type
head, head': A
tail': list A
d: A
IH: default d (last (head' :: tail')) =
nth (length (head' :: tail') - 1)
  (head' :: tail') d
default d
  match last (head' :: tail') with
  | Some _0 => Some _0
  | None => Some head
  end =
match length (head' :: tail') with
| 0 => head
| S _0 => nth _0 (head' :: tail') d
end rewrite -> (ltac:(simpl; reflexivity) : length (_ :: _) = _) in *.
A: Type
head, head': A
tail': list A
d: A
IH: default d (last (head' :: tail')) =
nth (S (length tail') - 1) (head' :: tail') d
default d
  match last (head' :: tail') with
  | Some _0 => Some _0
  | None => Some head
  end = nth (length tail') (head' :: tail') d rewrite (ltac:(lia) : S (length tail') - 1 = length tail') in IH.
A: Type
head, head': A
tail': list A
d: A
IH: default d (last (head' :: tail')) =
nth (length tail') (head' :: tail') d
default d
  match last (head' :: tail') with
  | Some _0 => Some _0
  | None => Some head
  end = nth (length tail') (head' :: tail') d rewrite <- IH.
A: Type
head, head': A
tail': list A
d: A
IH: default d (last (head' :: tail')) =
nth (length tail') (head' :: tail') d
default d
  match last (head' :: tail') with
  | Some _0 => Some _0
  | None => Some head
  end = default d (last (head' :: tail'))
  assert (h : last (head' :: tail') <> None).
A: Type
head, head': A
tail': list A
d: A
IH: default d (last (head' :: tail')) =
nth (length tail') (head' :: tail') d
last (head' :: tail') ≠ None
A: Type
head, head': A
tail': list A
d: A
IH: default d (last (head' :: tail')) =
nth (length tail') (head' :: tail') d
h: last (head' :: tail') ≠ None
default d
  match last (head' :: tail') with
  | Some _0 => Some _0
  | None => Some head
  end = default d (last (head' :: tail'))
  {
A: Type
head, head': A
tail': list A
d: A
IH: default d (last (head' :: tail')) =
nth (length tail') (head' :: tail') d
last (head' :: tail') ≠ None rewrite last_cons.
A: Type
head, head': A
tail': list A
d: A
IH: default d (last (head' :: tail')) =
nth (length tail') (head' :: tail') d
match last tail' with
| Some _0 => Some _0
| None => Some head'
end ≠ None destruct (last tail'); easy. }
A: Type
head, head': A
tail': list A
d: A
IH: default d (last (head' :: tail')) =
nth (length tail') (head' :: tail') d
h: last (head' :: tail') ≠ None
default d
  match last (head' :: tail') with
  | Some _0 => Some _0
  | None => Some head
  end = default d (last (head' :: tail'))
  destruct (last (head' :: tail')) as [x |].
A: Type
head, head': A
tail': list A
d, x: A
IH: default d (Some x) =
nth (length tail') (head' :: tail') d
h: Some x ≠ None
default d (Some x) = default d (Some x)
A: Type
head, head': A
tail': list A
d: A
IH: default d None =
nth (length tail') (head' :: tail') d
h: None ≠ None
default d (Some head) = default d None {
A: Type
head, head': A
tail': list A
d, x: A
IH: default d (Some x) =
nth (length tail') (head' :: tail') d
h: Some x ≠ None
default d (Some x) = default d (Some x) reflexivity. }
A: Type
head, head': A
tail': list A
d: A
IH: default d None =
nth (length tail') (head' :: tail') d
h: None ≠ None
default d (Some head) = default d None easy.
Qed.

Lemma ancestorEqLastAncestorChain dsu fuel index : nth (length (ancestorChain dsu fuel index) - 1) (ancestorChain dsu fuel index) 0 = ancestor dsu fuel index.
dsu: list Slot
fuel, index: nat
nth (length (ancestorChain dsu fuel index) - 1)
  (ancestorChain dsu fuel index) 0 =
ancestor dsu fuel index
Proof.
dsu: list Slot
fuel, index: nat
nth (length (ancestorChain dsu fuel index) - 1)
  (ancestorChain dsu fuel index) 0 =
ancestor dsu fuel index
  pose proof defaultLast (ancestorChain dsu fuel index) 0 as h.
dsu: list Slot
fuel, index: nat
h: default 0 (last (ancestorChain dsu fuel index)) =
nth (length (ancestorChain dsu fuel index) - 1)
  (ancestorChain dsu fuel index) 0
nth (length (ancestorChain dsu fuel index) - 1)
  (ancestorChain dsu fuel index) 0 =
ancestor dsu fuel index rewrite ancestorEqLastAncestorChain' in h.
dsu: list Slot
fuel, index: nat
h: default 0 (Some (ancestor dsu fuel index)) =
nth (length (ancestorChain dsu fuel index) - 1)
  (ancestorChain dsu fuel index) 0
nth (length (ancestorChain dsu fuel index) - 1)
  (ancestorChain dsu fuel index) 0 =
ancestor dsu fuel index simpl in h.
dsu: list Slot
fuel, index: nat
h: ancestor dsu fuel index =
nth (length (ancestorChain dsu fuel index) - 1)
  (ancestorChain dsu fuel index) 0
nth (length (ancestorChain dsu fuel index) - 1)
  (ancestorChain dsu fuel index) 0 =
ancestor dsu fuel index symmetry.
dsu: list Slot
fuel, index: nat
h: ancestor dsu fuel index =
nth (length (ancestorChain dsu fuel index) - 1)
  (ancestorChain dsu fuel index) 0
ancestor dsu fuel index =
nth (length (ancestorChain dsu fuel index) - 1)
  (ancestorChain dsu fuel index) 0 exact h.
Qed.

Definition validChain (dsu : list Slot) (chain : list nat) := chain <> [] /\ nth 0 chain 0 < length dsu /\ forall index, S index < length chain -> nth (nth index chain 0) dsu (Ancestor Unit) = ReferTo (nth (S index) chain 0).

Definition validChainToAncestor (dsu : list Slot) (chain : list nat) := validChain dsu chain /\ exists x, nth (nth (length chain - 1) chain 0) dsu (Ancestor Unit) = Ancestor x.

Lemma pigeonhole (f : nat -> nat) (n : nat) (hImage : forall x, f x < n) : exists i j, i <> j /\ i < S n /\ j < S n /\ f i = f j.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S n ∧ _1 < S n ∧ f _0 = f _1
Proof.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S n ∧ _1 < S n ∧ f _0 = f _1
  induction n as [| n IH] in f, hImage |- *.
f: nat → nat
hImage: ∀ _0 : nat, f _0 < 0
∃ _0 _1 : nat, _0 ≠ _1 ∧ _0 < 1 ∧ _1 < 1 ∧ f _0 = f _1
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1
  {
f: nat → nat
hImage: ∀ _0 : nat, f _0 < 0
∃ _0 _1 : nat, _0 ≠ _1 ∧ _0 < 1 ∧ _1 < 1 ∧ f _0 = f _1 pose proof hImage 0.
f: nat → nat
hImage: ∀ _0 : nat, f _0 < 0
_0: f 0 < 0
∃ _0 _1 : nat, _0 ≠ _1 ∧ _0 < 1 ∧ _1 < 1 ∧ f _0 = f _1 lia. }
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1
  remember (existsInRange (S n) (fun x => bool_decide (f x = f (S n)))) as s eqn:hs.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
s: bool
hs: s =
existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n)))
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 symmetry in hs.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
s: bool
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) = s
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 destruct s.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
true
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1
  -
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
true
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 pose proof proj1 (existsInRangeMeaning _ _) (Is_true_true_2 _ hs) as [w [h1 h2]].
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
true
w: nat
h1: w < S n
h2: bool_decide (f w = f (S n)) = true
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 rewrite bool_decide_eq_true in h2.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
true
w: nat
h1: w < S n
h2: f w = f (S n)
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 exists w.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
true
w: nat
h1: w < S n
h2: f w = f (S n)
∃ _0 : nat,
  w ≠ _0 ∧ w < S (S n) ∧ _0 < S (S n) ∧ f w = f _0 exists (S n).
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
true
w: nat
h1: w < S n
h2: f w = f (S n)
w ≠ S n ∧ w < S (S n) ∧ S n < S (S n) ∧ f w = f (S n) repeat split; try lia.
  -
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 pose proof proj1 (notExistsInRangeMeaning _ _) (Is_true_false_2 _ hs) as step.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: notExistsInRangeLogic (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n)))
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 unfold notExistsInRangeLogic in step.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 assert (step2 : forall x, x < S n -> f x <> f (S n)).
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 {
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
∀ _0 : nat, _0 < S n → f _0 ≠ f (S n) intros x h.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
x: nat
h: x < S n
f x ≠ f (S n) pose proof step x ltac:(lia) as h2.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
x: nat
h: x < S n
h2: ¬ bool_decide (f x = f (S n))
f x ≠ f (S n) case_bool_decide as h1.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
x: nat
h: x < S n
h2: ¬ true
h1: f x = f (S n)
f x ≠ f (S n)
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
x: nat
h: x < S n
h2: ¬ false
h1: f x ≠ f (S n)
f x ≠ f (S n) {
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
x: nat
h: x < S n
h2: ¬ true
h1: f x = f (S n)
f x ≠ f (S n) exfalso.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
x: nat
h: x < S n
h2: ¬ true
h1: f x = f (S n)
False apply h2.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
x: nat
h: x < S n
h2: ¬ true
h1: f x = f (S n)
true easy. }
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
x: nat
h: x < S n
h2: ¬ false
h1: f x ≠ f (S n)
f x ≠ f (S n) assumption. }
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1
    destruct (decide (n = 0)) as [h1 | h1].
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n = 0
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1
    {
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n = 0
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 pose proof hImage 0 as h2.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n = 0
h2: f 0 < S n
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 pose proof hImage 1 as h3.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n = 0
h2: f 0 < S n
h3: f 1 < S n
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 subst n.
f: nat → nat
step2: ∀ _0 : nat, _0 < 1 → f _0 ≠ f 1
step: ∀ _0 : nat, _0 < 1 → ¬ bool_decide (f _0 = f 1)
hs: existsInRange 1
  (λ _0 : nat, bool_decide (f _0 = f 1)) = false
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < 0)
  → ∃ _1 _2 : nat,
      _1 ≠ _2 ∧ _1 < 1 ∧ _2 < 1 ∧ _0 _1 = _0 _2
hImage: ∀ _0 : nat, f _0 < 1
h3: f 1 < 1
h2: f 0 < 1
∃ _0 _1 : nat, _0 ≠ _1 ∧ _0 < 2 ∧ _1 < 2 ∧ f _0 = f _1 exists 0, 1.
f: nat → nat
step2: ∀ _0 : nat, _0 < 1 → f _0 ≠ f 1
step: ∀ _0 : nat, _0 < 1 → ¬ bool_decide (f _0 = f 1)
hs: existsInRange 1
  (λ _0 : nat, bool_decide (f _0 = f 1)) = false
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < 0)
  → ∃ _1 _2 : nat,
      _1 ≠ _2 ∧ _1 < 1 ∧ _2 < 1 ∧ _0 _1 = _0 _2
hImage: ∀ _0 : nat, f _0 < 1
h3: f 1 < 1
h2: f 0 < 1
0 ≠ 1 ∧ 0 < 2 ∧ 1 < 2 ∧ f 0 = f 1 split; lia. }
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1
    pose proof IH (fun x => if decide (f x < f (S n)) then f x else f x - 1) ltac:(intro x; simpl; case_decide as h; [pose proof hImage (S n) as h'; lia | pose proof hImage x as h'; lia]) as [a [b [c [d [e g]]]]].
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
g: (if decide (f a < f (S n)) then f a else f a - 1) =
(if decide (f b < f (S n)) then f b else f b - 1)
∃ _0 _1 : nat,
  _0 ≠ _1 ∧ _0 < S (S n) ∧ _1 < S (S n) ∧ f _0 = f _1 exists a, b.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
g: (if decide (f a < f (S n)) then f a else f a - 1) =
(if decide (f b < f (S n)) then f b else f b - 1)
a ≠ b ∧ a < S (S n) ∧ b < S (S n) ∧ f a = f b repeat split; try (assumption || lia).
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
g: (if decide (f a < f (S n)) then f a else f a - 1) =
(if decide (f b < f (S n)) then f b else f b - 1)
f a = f b case_decide as h2; case_decide as h3; try assumption.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
h2: f a < f (S n)
h3: ¬ f b < f (S n)
g: f a = f b - 1
f a = f b
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
h2: ¬ f a < f (S n)
h3: f b < f (S n)
g: f a - 1 = f b
f a = f b
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
h2: ¬ f a < f (S n)
h3: ¬ f b < f (S n)
g: f a - 1 = f b - 1
f a = f b
    +
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
h2: f a < f (S n)
h3: ¬ f b < f (S n)
g: f a = f b - 1
f a = f b pose proof step2 b e.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
h2: f a < f (S n)
h3: ¬ f b < f (S n)
g: f a = f b - 1
_0: f b ≠ f (S n)
f a = f b lia.
    +
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
h2: ¬ f a < f (S n)
h3: f b < f (S n)
g: f a - 1 = f b
f a = f b pose proof step2 a d.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
h2: ¬ f a < f (S n)
h3: f b < f (S n)
g: f a - 1 = f b
_0: f a ≠ f (S n)
f a = f b lia.
    +
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
h2: ¬ f a < f (S n)
h3: ¬ f b < f (S n)
g: f a - 1 = f b - 1
f a = f b pose proof step2 b e.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
h2: ¬ f a < f (S n)
h3: ¬ f b < f (S n)
g: f a - 1 = f b - 1
_0: f b ≠ f (S n)
f a = f b pose proof step2 a d.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < S n
IH: ∀ _0 : nat → nat,
  (∀ _1 : nat, _0 _1 < n)
  → ∃ _1 _2 : nat,
      _1 ≠ _2
      ∧ _1 < S n ∧ _2 < S n ∧ _0 _1 = _0 _2
hs: existsInRange (S n)
  (λ _0 : nat, bool_decide (f _0 = f (S n))) =
false
step: ∀ _0 : nat,
  _0 < S n → ¬ bool_decide (f _0 = f (S n))
step2: ∀ _0 : nat, _0 < S n → f _0 ≠ f (S n)
h1: n ≠ 0
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
h2: ¬ f a < f (S n)
h3: ¬ f b < f (S n)
g: f a - 1 = f b - 1
_0: f b ≠ f (S n)
_1: f a ≠ f (S n)
f a = f b lia.
Qed.

Lemma pigeonholeOrdered (f : nat -> nat) (n : nat) (hImage : forall x, f x < n) : exists i j, i < j /\ i < S n /\ j < S n /\ f i = f j.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
∃ _0 _1 : nat,
  _0 < _1 ∧ _0 < S n ∧ _1 < S n ∧ f _0 = f _1
Proof.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
∃ _0 _1 : nat,
  _0 < _1 ∧ _0 < S n ∧ _1 < S n ∧ f _0 = f _1
  pose proof pigeonhole f n hImage as [a [b [c [d [e g]]]]].
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
g: f a = f b
∃ _0 _1 : nat,
  _0 < _1 ∧ _0 < S n ∧ _1 < S n ∧ f _0 = f _1
  destruct (ltac:(lia) : a < b \/ b < a) as [h | h].
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
g: f a = f b
h: a < b
∃ _0 _1 : nat,
  _0 < _1 ∧ _0 < S n ∧ _1 < S n ∧ f _0 = f _1
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
g: f a = f b
h: b < a
∃ _0 _1 : nat,
  _0 < _1 ∧ _0 < S n ∧ _1 < S n ∧ f _0 = f _1
  -
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
g: f a = f b
h: a < b
∃ _0 _1 : nat,
  _0 < _1 ∧ _0 < S n ∧ _1 < S n ∧ f _0 = f _1 exists a, b.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
g: f a = f b
h: a < b
a < b ∧ a < S n ∧ b < S n ∧ f a = f b repeat split; assumption.
  -
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
g: f a = f b
h: b < a
∃ _0 _1 : nat,
  _0 < _1 ∧ _0 < S n ∧ _1 < S n ∧ f _0 = f _1 exists b, a.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
g: f a = f b
h: b < a
b < a ∧ b < S n ∧ a < S n ∧ f b = f a symmetry in g.
f: nat → nat
n: nat
hImage: ∀ _0 : nat, f _0 < n
a, b: nat
c: a ≠ b
d: a < S n
e: b < S n
g: f b = f a
h: b < a
b < a ∧ b < S n ∧ a < S n ∧ f b = f a repeat split; assumption.
Qed.

Lemma commonSubarrays (dsu : list Slot) chain (h2 : validChain dsu chain) (i j delta : nat) (hI : i < length chain) (hJ : j < length chain) (hIDelta : i + delta < length chain) (hJDelta : j + delta < length chain) (hEq : nth i chain 0 = nth j chain 0) : nth (i + delta) chain 0 = nth (j + delta) chain 0.
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + delta < length chain
hJDelta: j + delta < length chain
hEq: nth i chain 0 = nth j chain 0
nth (i + delta) chain 0 = nth (j + delta) chain 0
Proof.
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + delta < length chain
hJDelta: j + delta < length chain
hEq: nth i chain 0 = nth j chain 0
nth (i + delta) chain 0 = nth (j + delta) chain 0
  induction delta as [| delta IH].
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + 0 < length chain
hJDelta: j + 0 < length chain
hEq: nth i chain 0 = nth j chain 0
nth (i + 0) chain 0 = nth (j + 0) chain 0
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
nth (i + S delta) chain 0 = nth (j + S delta) chain 0
  {
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + 0 < length chain
hJDelta: j + 0 < length chain
hEq: nth i chain 0 = nth j chain 0
nth (i + 0) chain 0 = nth (j + 0) chain 0 now rewrite !Nat.add_0_r. }
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
nth (i + S delta) chain 0 = nth (j + S delta) chain 0
  pose proof IH ltac:(lia) ltac:(lia) as h.
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
h: nth (i + delta) chain 0 = nth (j + delta) chain 0
nth (i + S delta) chain 0 = nth (j + S delta) chain 0
  pose proof h2 as [a [b c]].
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
h: nth (i + delta) chain 0 = nth (j + delta) chain 0
a: chain ≠ []
b: nth 0 chain 0 < length dsu
c: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
nth (i + S delta) chain 0 = nth (j + S delta) chain 0
  pose proof c (i + delta) ltac:(lia) as c1.
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
h: nth (i + delta) chain 0 = nth (j + delta) chain 0
a: chain ≠ []
b: nth 0 chain 0 < length dsu
c: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
c1: nth (nth (i + delta) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (i + delta)) chain 0)
nth (i + S delta) chain 0 = nth (j + S delta) chain 0 pose proof c (j + delta) ltac:(lia) as c2.
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
h: nth (i + delta) chain 0 = nth (j + delta) chain 0
a: chain ≠ []
b: nth 0 chain 0 < length dsu
c: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
c1: nth (nth (i + delta) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (i + delta)) chain 0)
c2: nth (nth (j + delta) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (j + delta)) chain 0)
nth (i + S delta) chain 0 = nth (j + S delta) chain 0 rewrite h in c1.
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
h: nth (i + delta) chain 0 = nth (j + delta) chain 0
a: chain ≠ []
b: nth 0 chain 0 < length dsu
c: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
c1: nth (nth (j + delta) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (i + delta)) chain 0)
c2: nth (nth (j + delta) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (j + delta)) chain 0)
nth (i + S delta) chain 0 = nth (j + S delta) chain 0 symmetry in c1.
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
h: nth (i + delta) chain 0 = nth (j + delta) chain 0
a: chain ≠ []
b: nth 0 chain 0 < length dsu
c: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
c1: ReferTo (nth (S (i + delta)) chain 0) =
nth (nth (j + delta) chain 0) dsu (Ancestor Unit)
c2: nth (nth (j + delta) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (j + delta)) chain 0)
nth (i + S delta) chain 0 = nth (j + S delta) chain 0 pose proof eq_trans c1 c2 as c3.
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
h: nth (i + delta) chain 0 = nth (j + delta) chain 0
a: chain ≠ []
b: nth 0 chain 0 < length dsu
c: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
c1: ReferTo (nth (S (i + delta)) chain 0) =
nth (nth (j + delta) chain 0) dsu (Ancestor Unit)
c2: nth (nth (j + delta) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (j + delta)) chain 0)
c3: ReferTo (nth (S (i + delta)) chain 0) =
ReferTo (nth (S (j + delta)) chain 0)
nth (i + S delta) chain 0 = nth (j + S delta) chain 0 injection c3.
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
h: nth (i + delta) chain 0 = nth (j + delta) chain 0
a: chain ≠ []
b: nth 0 chain 0 < length dsu
c: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
c1: ReferTo (nth (S (i + delta)) chain 0) =
nth (nth (j + delta) chain 0) dsu (Ancestor Unit)
c2: nth (nth (j + delta) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (j + delta)) chain 0)
c3: ReferTo (nth (S (i + delta)) chain 0) =
ReferTo (nth (S (j + delta)) chain 0)
nth (S (i + delta)) chain 0 =
nth (S (j + delta)) chain 0
→ nth (i + S delta) chain 0 =
  nth (j + S delta) chain 0 rewrite !Nat.add_succ_r.
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
h: nth (i + delta) chain 0 = nth (j + delta) chain 0
a: chain ≠ []
b: nth 0 chain 0 < length dsu
c: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
c1: ReferTo (nth (S (i + delta)) chain 0) =
nth (nth (j + delta) chain 0) dsu (Ancestor Unit)
c2: nth (nth (j + delta) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (j + delta)) chain 0)
c3: ReferTo (nth (S (i + delta)) chain 0) =
ReferTo (nth (S (j + delta)) chain 0)
nth (S (i + delta)) chain 0 =
nth (S (j + delta)) chain 0
→ nth (S (i + delta)) chain 0 =
  nth (S (j + delta)) chain 0 intro h3.
dsu: list Slot
chain: list nat
h2: validChain dsu chain
i, j, delta: nat
hI: i < length chain
hJ: j < length chain
hIDelta: i + S delta < length chain
hJDelta: j + S delta < length chain
hEq: nth i chain 0 = nth j chain 0
IH: i + delta < length chain
→ j + delta < length chain
  → nth (i + delta) chain 0 =
    nth (j + delta) chain 0
h: nth (i + delta) chain 0 = nth (j + delta) chain 0
a: chain ≠ []
b: nth 0 chain 0 < length dsu
c: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
c1: ReferTo (nth (S (i + delta)) chain 0) =
nth (nth (j + delta) chain 0) dsu (Ancestor Unit)
c2: nth (nth (j + delta) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (j + delta)) chain 0)
c3: ReferTo (nth (S (i + delta)) chain 0) =
ReferTo (nth (S (j + delta)) chain 0)
h3: nth (S (i + delta)) chain 0 =
nth (S (j + delta)) chain 0
nth (S (i + delta)) chain 0 =
nth (S (j + delta)) chain 0 exact h3.
Qed.

Lemma chainElementsValidIndices (dsu : list Slot) chain (h1 : noIllegalIndices dsu) (h2 : validChainToAncestor dsu chain) index (h3 : index < length chain) : nth index chain 0 < length dsu.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
index: nat
h3: index < length chain
nth index chain 0 < length dsu
Proof.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
index: nat
h3: index < length chain
nth index chain 0 < length dsu
  induction index as [| index IH].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h3: 0 < length chain
nth 0 chain 0 < length dsu
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
index: nat
h3: S index < length chain
IH: index < length chain
→ nth index chain 0 < length dsu
nth (S index) chain 0 < length dsu
  -
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h3: 0 < length chain
nth 0 chain 0 < length dsu pose proof h2 as [[_ [x _]] _].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h3: 0 < length chain
x: nth 0 chain 0 < length dsu
nth 0 chain 0 < length dsu exact x.
  -
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
index: nat
h3: S index < length chain
IH: index < length chain
→ nth index chain 0 < length dsu
nth (S index) chain 0 < length dsu apply (h1 (nth index chain 0) (nth (S index) chain 0)).
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
index: nat
h3: S index < length chain
IH: index < length chain
→ nth index chain 0 < length dsu
nth (nth index chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S index) chain 0) pose proof h2 as [[_ [_ x]] _].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
index: nat
h3: S index < length chain
IH: index < length chain
→ nth index chain 0 < length dsu
x: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
nth (nth index chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S index) chain 0) exact (x index h3).
Qed.

Lemma validChainPairwiseDifferent (dsu : list Slot) chain (h1 : noIllegalIndices dsu) (h2 : validChainToAncestor dsu chain) (i j : nat) (h4 : i <> j) (h5 : i < length chain) (h3 : j < length chain) : nth i chain 0 <> nth j chain 0.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
nth i chain 0 ≠ nth j chain 0
Proof.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
nth i chain 0 ≠ nth j chain 0
  assert (wlog : forall i j, i < j -> i < length chain -> j < length chain -> nth i chain 0 <> nth j chain 0).
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
∀ _0 _1 : nat,
  _0 < _1
  → _0 < length chain
    → _1 < length chain
      → nth _0 chain 0 ≠ nth _1 chain 0
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
wlog: ∀ _0 _1 : nat,
  _0 < _1
  → _0 < length chain
    → _1 < length chain
      → nth _0 chain 0 ≠ nth _1 chain 0
nth i chain 0 ≠ nth j chain 0
  {
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
∀ _0 _1 : nat,
  _0 < _1
  → _0 < length chain
    → _1 < length chain
      → nth _0 chain 0 ≠ nth _1 chain 0 revert h1 h2.
dsu: list Slot
chain: list nat
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
noIllegalIndices dsu
→ validChainToAncestor dsu chain
  → ∀ _0 _1 : nat,
      _0 < _1
      → _0 < length chain
        → _1 < length chain
          → nth _0 chain 0 ≠ nth _1 chain 0 clear.
dsu: list Slot
chain: list nat
noIllegalIndices dsu
→ validChainToAncestor dsu chain
  → ∀ _0 _1 : nat,
      _0 < _1
      → _0 < length chain
        → _1 < length chain
          → nth _0 chain 0 ≠ nth _1 chain 0 intros h1 h2 i j h3 h4 h5 h6.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h3: i < j
h4: i < length chain
h5: j < length chain
h6: nth i chain 0 = nth j chain 0
False
    destruct h2 as [[e [f g]] [b c]].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
i, j: nat
h3: i < j
h4: i < length chain
h5: j < length chain
h6: nth i chain 0 = nth j chain 0
False
    pose proof commonSubarrays dsu chain ltac:(repeat split; assumption) i j (length chain - 1 - j) h4 h5 ltac:(lia) ltac:(lia) h6 as step.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
i, j: nat
h3: i < j
h4: i < length chain
h5: j < length chain
h6: nth i chain 0 = nth j chain 0
step: nth (i + (length chain - 1 - j)) chain 0 =
nth (j + (length chain - 1 - j)) chain 0
False pose proof (ltac:(destruct chain; simpl in *; lia) : 0 < length chain).
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
i, j: nat
h3: i < j
h4: i < length chain
h5: j < length chain
h6: nth i chain 0 = nth j chain 0
step: nth (i + (length chain - 1 - j)) chain 0 =
nth (j + (length chain - 1 - j)) chain 0
_0: 0 < length chain
False
    rewrite (ltac:(lia) : j + (length chain - 1 - j) = length chain - 1) in step.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
i, j: nat
h3: i < j
h4: i < length chain
h5: j < length chain
h6: nth i chain 0 = nth j chain 0
step: nth (i + (length chain - 1 - j)) chain 0 =
nth (length chain - 1) chain 0
_0: 0 < length chain
False rewrite <- step in c.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
i, j: nat
c: nth (nth (i + (length chain - 1 - j)) chain 0) dsu
  (Ancestor Unit) = Ancestor b
h3: i < j
h4: i < length chain
h5: j < length chain
h6: nth i chain 0 = nth j chain 0
step: nth (i + (length chain - 1 - j)) chain 0 =
nth (length chain - 1) chain 0
_0: 0 < length chain
False
    rewrite (g (i + (length chain - 1 - j)) ltac:(lia)) in c.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
i, j: nat
c: ReferTo
  (nth (S (i + (length chain - 1 - j))) chain 0) =
Ancestor b
h3: i < j
h4: i < length chain
h5: j < length chain
h6: nth i chain 0 = nth j chain 0
step: nth (i + (length chain - 1 - j)) chain 0 =
nth (length chain - 1) chain 0
_0: 0 < length chain
False easy. }
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
wlog: ∀ _0 _1 : nat,
  _0 < _1
  → _0 < length chain
    → _1 < length chain
      → nth _0 chain 0 ≠ nth _1 chain 0
nth i chain 0 ≠ nth j chain 0
  destruct (ltac:(lia) : i < j \/ j < i) as [h | h].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
wlog: ∀ _0 _1 : nat,
  _0 < _1
  → _0 < length chain
    → _1 < length chain
      → nth _0 chain 0 ≠ nth _1 chain 0
h: i < j
nth i chain 0 ≠ nth j chain 0
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
wlog: ∀ _0 _1 : nat,
  _0 < _1
  → _0 < length chain
    → _1 < length chain
      → nth _0 chain 0 ≠ nth _1 chain 0
h: j < i
nth i chain 0 ≠ nth j chain 0
  -
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
wlog: ∀ _0 _1 : nat,
  _0 < _1
  → _0 < length chain
    → _1 < length chain
      → nth _0 chain 0 ≠ nth _1 chain 0
h: i < j
nth i chain 0 ≠ nth j chain 0 apply wlog; assumption.
  -
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
wlog: ∀ _0 _1 : nat,
  _0 < _1
  → _0 < length chain
    → _1 < length chain
      → nth _0 chain 0 ≠ nth _1 chain 0
h: j < i
nth i chain 0 ≠ nth j chain 0 intro g.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
wlog: ∀ _0 _1 : nat,
  _0 < _1
  → _0 < length chain
    → _1 < length chain
      → nth _0 chain 0 ≠ nth _1 chain 0
h: j < i
g: nth i chain 0 = nth j chain 0
False symmetry in g.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
wlog: ∀ _0 _1 : nat,
  _0 < _1
  → _0 < length chain
    → _1 < length chain
      → nth _0 chain 0 ≠ nth _1 chain 0
h: j < i
g: nth j chain 0 = nth i chain 0
False revert g.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
i, j: nat
h4: i ≠ j
h5: i < length chain
h3: j < length chain
wlog: ∀ _0 _1 : nat,
  _0 < _1
  → _0 < length chain
    → _1 < length chain
      → nth _0 chain 0 ≠ nth _1 chain 0
h: j < i
nth j chain 0 = nth i chain 0 → False apply wlog; assumption.
Qed.

Lemma validChainMaxLength (dsu : list Slot) chain (h1 : noIllegalIndices dsu) (h2 : validChainToAncestor dsu chain) : length chain <= length dsu.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
length chain ≤ length dsu
Proof.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
length chain ≤ length dsu
  assert (h : forall index, index < length chain -> nth index chain 0 < length dsu).
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
length chain ≤ length dsu
  {
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu apply chainElementsValidIndices; assumption. }
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
length chain ≤ length dsu
  destruct (ltac:(lia) : length dsu = 0 \/ 0 < length dsu) as [hs | hs].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: length dsu = 0
length chain ≤ length dsu
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
length chain ≤ length dsu
  {
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: length dsu = 0
length chain ≤ length dsu destruct h2 as [[e [f g]] [b c]].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: length dsu = 0
length chain ≤ length dsu lia. }
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
length chain ≤ length dsu
  assert (h3 : forall x : nat, nth x chain 0 < length dsu).
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
∀ _0 : nat, nth _0 chain 0 < length dsu
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
length chain ≤ length dsu
  {
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
∀ _0 : nat, nth _0 chain 0 < length dsu intro x.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
x: nat
nth x chain 0 < length dsu destruct (ltac:(lia) : length chain <= x \/ x < length chain) as [hd | hd].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
x: nat
hd: length chain ≤ x
nth x chain 0 < length dsu
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
x: nat
hd: x < length chain
nth x chain 0 < length dsu {
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
x: nat
hd: length chain ≤ x
nth x chain 0 < length dsu rewrite nth_overflow; lia. }
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
x: nat
hd: x < length chain
nth x chain 0 < length dsu apply h.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
x: nat
hd: x < length chain
x < length chain exact hd. }
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
length chain ≤ length dsu
  pose proof pigeonholeOrdered (fun x => nth x chain 0) (length dsu) h3 as [a [b [c [d [e f]]]]].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
a, b: nat
c: a < b
d: a < S (length dsu)
e: b < S (length dsu)
f: nth a chain 0 = nth b chain 0
length chain ≤ length dsu
  pose proof commonSubarrays dsu chain ltac:(pose proof h2 as [h4 _]; exact h4) a b (length chain - 1 - b) as g.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
a, b: nat
c: a < b
d: a < S (length dsu)
e: b < S (length dsu)
f: nth a chain 0 = nth b chain 0
g: a < length chain
→ b < length chain
  → a + (length chain - 1 - b) < length chain
    → b + (length chain - 1 - b) < length chain
      → nth a chain 0 = nth b chain 0
        → nth (a + (length chain - 1 - b)) chain 0 =
          nth (b + (length chain - 1 - b)) chain 0
length chain ≤ length dsu
  pose proof (ltac:(lia) : length chain <= length dsu \/ length dsu < length chain) as [hd | hd]; [exact hd |].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
a, b: nat
c: a < b
d: a < S (length dsu)
e: b < S (length dsu)
f: nth a chain 0 = nth b chain 0
g: a < length chain
→ b < length chain
  → a + (length chain - 1 - b) < length chain
    → b + (length chain - 1 - b) < length chain
      → nth a chain 0 = nth b chain 0
        → nth (a + (length chain - 1 - b)) chain 0 =
          nth (b + (length chain - 1 - b)) chain 0
hd: length dsu < length chain
length chain ≤ length dsu
  pose proof g ltac:(lia) ltac:(lia) ltac:(lia) ltac:(lia) f as i.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
a, b: nat
c: a < b
d: a < S (length dsu)
e: b < S (length dsu)
f: nth a chain 0 = nth b chain 0
g: a < length chain
→ b < length chain
  → a + (length chain - 1 - b) < length chain
    → b + (length chain - 1 - b) < length chain
      → nth a chain 0 = nth b chain 0
        → nth (a + (length chain - 1 - b)) chain 0 =
          nth (b + (length chain - 1 - b)) chain 0
hd: length dsu < length chain
i: nth (a + (length chain - 1 - b)) chain 0 =
nth (b + (length chain - 1 - b)) chain 0
length chain ≤ length dsu clear g.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
a, b: nat
c: a < b
d: a < S (length dsu)
e: b < S (length dsu)
f: nth a chain 0 = nth b chain 0
hd: length dsu < length chain
i: nth (a + (length chain - 1 - b)) chain 0 =
nth (b + (length chain - 1 - b)) chain 0
length chain ≤ length dsu rewrite (ltac:(lia) : b + (length chain - 1 - b) = length chain - 1) in i.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
a, b: nat
c: a < b
d: a < S (length dsu)
e: b < S (length dsu)
f: nth a chain 0 = nth b chain 0
hd: length dsu < length chain
i: nth (a + (length chain - 1 - b)) chain 0 =
nth (length chain - 1) chain 0
length chain ≤ length dsu
  pose proof h2 as [[_ [_ g]] _].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
a, b: nat
c: a < b
d: a < S (length dsu)
e: b < S (length dsu)
f: nth a chain 0 = nth b chain 0
hd: length dsu < length chain
i: nth (a + (length chain - 1 - b)) chain 0 =
nth (length chain - 1) chain 0
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
length chain ≤ length dsu pose proof g (a + (length chain - 1 - b)) ltac:(lia) as g1.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
a, b: nat
c: a < b
d: a < S (length dsu)
e: b < S (length dsu)
f: nth a chain 0 = nth b chain 0
hd: length dsu < length chain
i: nth (a + (length chain - 1 - b)) chain 0 =
nth (length chain - 1) chain 0
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
g1: nth (nth (a + (length chain - 1 - b)) chain 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S (a + (length chain - 1 - b))) chain 0)
length chain ≤ length dsu rewrite i in g1.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
a, b: nat
c: a < b
d: a < S (length dsu)
e: b < S (length dsu)
f: nth a chain 0 = nth b chain 0
hd: length dsu < length chain
i: nth (a + (length chain - 1 - b)) chain 0 =
nth (length chain - 1) chain 0
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
g1: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) =
ReferTo
  (nth (S (a + (length chain - 1 - b))) chain 0)
length chain ≤ length dsu pose proof h2 as [_ [x g2]].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
a, b: nat
c: a < b
d: a < S (length dsu)
e: b < S (length dsu)
f: nth a chain 0 = nth b chain 0
hd: length dsu < length chain
i: nth (a + (length chain - 1 - b)) chain 0 =
nth (length chain - 1) chain 0
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
g1: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) =
ReferTo
  (nth (S (a + (length chain - 1 - b))) chain 0)
x: Tree
g2: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor x
length chain ≤ length dsu rewrite g1 in g2.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
h: ∀ _0 : nat,
  _0 < length chain → nth _0 chain 0 < length dsu
hs: 0 < length dsu
h3: ∀ _0 : nat, nth _0 chain 0 < length dsu
a, b: nat
c: a < b
d: a < S (length dsu)
e: b < S (length dsu)
f: nth a chain 0 = nth b chain 0
hd: length dsu < length chain
i: nth (a + (length chain - 1 - b)) chain 0 =
nth (length chain - 1) chain 0
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
g1: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) =
ReferTo
  (nth (S (a + (length chain - 1 - b))) chain 0)
x: Tree
g2: ReferTo
  (nth (S (a + (length chain - 1 - b))) chain 0) =
Ancestor x
length chain ≤ length dsu easy.
Qed.

Lemma validChainLe (dsu : list Slot) chain (h1 : noIllegalIndices dsu) (h2 : validChainToAncestor dsu chain) fuel vertex (h3 : nth 0 chain 0 = vertex) (h4 : length chain <= fuel) : chain = ancestorChain dsu fuel vertex.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
h4: length chain ≤ fuel
chain = ancestorChain dsu fuel vertex
Proof.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
h4: length chain ≤ fuel
chain = ancestorChain dsu fuel vertex
  revert vertex h3 fuel h4.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
∀ _0 : nat,
  nth 0 chain 0 = _0
  → ∀ _1 : nat,
      length chain ≤ _1
      → chain = ancestorChain dsu _1 _0
  induction chain as [| head tail IH]; intros vertex h3 fuel h4.
dsu: list Slot
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu []
vertex: nat
h3: nth 0 [] 0 = vertex
fuel: nat
h4: length [] ≤ fuel
[] = ancestorChain dsu fuel vertex
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ fuel
head :: tail = ancestorChain dsu fuel vertex
  -
dsu: list Slot
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu []
vertex: nat
h3: nth 0 [] 0 = vertex
fuel: nat
h4: length [] ≤ fuel
[] = ancestorChain dsu fuel vertex pose proof h2 as [[e [f g]] [h i]].
dsu: list Slot
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu []
vertex: nat
h3: nth 0 [] 0 = vertex
fuel: nat
h4: length [] ≤ fuel
e: [] ≠ []
f: nth 0 [] 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length []
  → nth (nth _0 [] 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) [] 0)
h: Tree
i: nth (nth (length [] - 1) [] 0) dsu (Ancestor Unit) =
Ancestor h
[] = ancestorChain dsu fuel vertex exfalso.
dsu: list Slot
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu []
vertex: nat
h3: nth 0 [] 0 = vertex
fuel: nat
h4: length [] ≤ fuel
e: [] ≠ []
f: nth 0 [] 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length []
  → nth (nth _0 [] 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) [] 0)
h: Tree
i: nth (nth (length [] - 1) [] 0) dsu (Ancestor Unit) =
Ancestor h
False exact (e ltac:(reflexivity)).
  -
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ fuel
head :: tail = ancestorChain dsu fuel vertex pose proof h2 as [[e [f g]] [h i]].
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
head :: tail = ancestorChain dsu fuel vertex destruct fuel as [| fuel].
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
h4: length (head :: tail) ≤ 0
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
head :: tail = ancestorChain dsu 0 vertex
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ S fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
head :: tail = ancestorChain dsu (S fuel) vertex {
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
h4: length (head :: tail) ≤ 0
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
head :: tail = ancestorChain dsu 0 vertex simpl in h4.
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
h4: S (length tail) ≤ 0
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
head :: tail = ancestorChain dsu 0 vertex lia. }
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ S fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
head :: tail = ancestorChain dsu (S fuel) vertex simpl.
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ S fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
head :: tail =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end
    destruct (decide (length tail = 0)) as [hs | hs].
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ S fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail = 0
head :: tail =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ S fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
head :: tail =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end
    +
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ S fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail = 0
head :: tail =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end pose proof nil_length_inv _ hs.
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ S fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail = 0
_0: tail = []
head :: tail =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end subst tail.
dsu: list Slot
head: nat
h1: noIllegalIndices dsu
IH: validChainToAncestor dsu []
→ ∀ _0 : nat,
    nth 0 [] 0 = _0
    → ∀ _1 : nat,
        length [] ≤ _1
        → [] = ancestorChain dsu _1 _0
h2: validChainToAncestor dsu [head]
vertex: nat
h3: nth 0 [head] 0 = vertex
fuel: nat
g: ∀ _0 : nat,
  S _0 < length [head]
  → nth (nth _0 [head] 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) [head] 0)
f: nth 0 [head] 0 < length dsu
e: [head] ≠ []
h4: length [head] ≤ S fuel
h: Tree
hs: length [] = 0
i: nth (nth (length [head] - 1) [head] 0) dsu
  (Ancestor Unit) = Ancestor h
[head] =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end simpl in *.
dsu: list Slot
head: nat
h1: noIllegalIndices dsu
IH: validChainToAncestor dsu []
→ ∀ _0 : nat,
    0 = _0
    → ∀ _1 : nat,
        0 ≤ _1 → [] = ancestorChain dsu _1 _0
h2: validChainToAncestor dsu [head]
vertex: nat
h3: head = vertex
fuel: nat
g: ∀ _0 : nat,
  S _0 < 1
  → nth match _0 with
        | 0 => head
        | _ => 0
        end dsu (Ancestor Unit) =
    ReferTo match _0 with
            | 0 | _ => 0
            end
f: head < length dsu
e: [head] ≠ []
h4: 1 ≤ S fuel
h: Tree
hs: 0 = 0
i: nth head dsu (Ancestor Unit) = Ancestor h
[head] =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end subst head.
dsu: list Slot
h1: noIllegalIndices dsu
IH: validChainToAncestor dsu []
→ ∀ _0 : nat,
    0 = _0
    → ∀ _1 : nat,
        0 ≤ _1 → [] = ancestorChain dsu _1 _0
vertex: nat
h2: validChainToAncestor dsu [vertex]
fuel: nat
e: [vertex] ≠ []
f: vertex < length dsu
g: ∀ _0 : nat,
  S _0 < 1
  → nth match _0 with
        | 0 => vertex
        | _ => 0
        end dsu (Ancestor Unit) =
    ReferTo match _0 with
            | 0 | _ => 0
            end
h4: 1 ≤ S fuel
h: Tree
hs: 0 = 0
i: nth vertex dsu (Ancestor Unit) = Ancestor h
[vertex] =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end rewrite i.
dsu: list Slot
h1: noIllegalIndices dsu
IH: validChainToAncestor dsu []
→ ∀ _0 : nat,
    0 = _0
    → ∀ _1 : nat,
        0 ≤ _1 → [] = ancestorChain dsu _1 _0
vertex: nat
h2: validChainToAncestor dsu [vertex]
fuel: nat
e: [vertex] ≠ []
f: vertex < length dsu
g: ∀ _0 : nat,
  S _0 < 1
  → nth match _0 with
        | 0 => vertex
        | _ => 0
        end dsu (Ancestor Unit) =
    ReferTo match _0 with
            | 0 | _ => 0
            end
h4: 1 ≤ S fuel
h: Tree
hs: 0 = 0
i: nth vertex dsu (Ancestor Unit) = Ancestor h
[vertex] = [vertex] reflexivity.
    +
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ S fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
head :: tail =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end pose proof g 0 ltac:(simpl in *; lia) as step.
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ S fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth (nth 0 (head :: tail) 0) dsu
  (Ancestor Unit) =
ReferTo (nth 1 (head :: tail) 0)
head :: tail =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end simpl in step.
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
fuel: nat
h4: length (head :: tail) ≤ S fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth head dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
head :: tail =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end simpl in h3.
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
vertex: nat
h3: head = vertex
fuel: nat
h4: length (head :: tail) ≤ S fuel
e: head :: tail ≠ []
f: nth 0 (head :: tail) 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length (head :: tail)
  → nth (nth _0 (head :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (head :: tail) 0)
h: Tree
i: nth
  (nth (length (head :: tail) - 1) 
     (head :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth head dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
head :: tail =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end subst head.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
vertex :: tail =
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end rewrite step.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
vertex :: tail =
vertex :: ancestorChain dsu fuel (nth 0 tail 0) pose proof (fun x => IH x (nth 0 tail 0) ltac:(reflexivity) fuel ltac:(simpl in *; lia)) as step2.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
vertex :: tail =
vertex :: ancestorChain dsu fuel (nth 0 tail 0)
      assert (step3 : validChainToAncestor dsu tail).
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
validChainToAncestor dsu tail
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
step3: validChainToAncestor dsu tail
vertex :: tail =
vertex :: ancestorChain dsu fuel (nth 0 tail 0)
      {
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
validChainToAncestor dsu tail repeat split.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
tail ≠ []
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
nth 0 tail 0 < length dsu
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
∀ _0 : nat,
  S _0 < length tail
  → nth (nth _0 tail 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) tail 0)
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0 {
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
tail ≠ [] intro step3.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
step3: tail = []
False subst tail.
dsu: list Slot
h1: noIllegalIndices dsu
vertex: nat
IH: validChainToAncestor dsu []
→ ∀ _0 : nat,
    nth 0 [] 0 = _0
    → ∀ _1 : nat,
        length [] ≤ _1
        → [] = ancestorChain dsu _1 _0
h2: validChainToAncestor dsu [vertex]
fuel: nat
h4: length [vertex] ≤ S fuel
e: [vertex] ≠ []
f: nth 0 [vertex] 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length [vertex]
  → nth (nth _0 [vertex] 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) [vertex] 0)
h: Tree
step2: validChainToAncestor dsu []
→ [] = ancestorChain dsu fuel (nth 0 [] 0)
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 [] 0)
hs: length [] ≠ 0
i: nth (nth (length [vertex] - 1) [vertex] 0) dsu
  (Ancestor Unit) = Ancestor h
False exact (hs ltac:(reflexivity)). }
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
nth 0 tail 0 < length dsu
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
∀ _0 : nat,
  S _0 < length tail
  → nth (nth _0 tail 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) tail 0)
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0 {
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
nth 0 tail 0 < length dsu pose proof chainElementsValidIndices dsu (vertex :: tail) h1 h2 1 ltac:(simpl; lia) as step3.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
step3: nth 1 (vertex :: tail) 0 < length dsu
nth 0 tail 0 < length dsu simpl in step3.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
step3: nth 0 tail 0 < length dsu
nth 0 tail 0 < length dsu exact step3. }
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
∀ _0 : nat,
  S _0 < length tail
  → nth (nth _0 tail 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) tail 0)
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0 {
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
∀ _0 : nat,
  S _0 < length tail
  → nth (nth _0 tail 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) tail 0) intros a ha.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
a: nat
ha: S a < length tail
nth (nth a tail 0) dsu (Ancestor Unit) =
ReferTo (nth (S a) tail 0) pose proof g (S a) ltac:(simpl; lia) as step3.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
a: nat
ha: S a < length tail
step3: nth (nth (S a) (vertex :: tail) 0) dsu
  (Ancestor Unit) =
ReferTo (nth (S (S a)) (vertex :: tail) 0)
nth (nth a tail 0) dsu (Ancestor Unit) =
ReferTo (nth (S a) tail 0) simpl in step3.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
a: nat
ha: S a < length tail
step3: nth (nth a tail 0) dsu (Ancestor Unit) =
ReferTo (nth (S a) tail 0)
nth (nth a tail 0) dsu (Ancestor Unit) =
ReferTo (nth (S a) tail 0) exact step3. }
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0 {
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0 exists h.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
nth (nth (length tail - 1) tail 0) dsu (Ancestor Unit) =
Ancestor h rewrite (ltac:(simpl; lia) : length (vertex :: tail) - 1 = S (length tail - 1)) in i.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth (nth (S (length tail - 1)) (vertex :: tail) 0)
  dsu (Ancestor Unit) = 
Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
nth (nth (length tail - 1) tail 0) dsu (Ancestor Unit) =
Ancestor h rewrite (ltac:(intros; simpl; reflexivity) : forall a b c d, nth (S a) (b :: c) d = nth a c d) in i.
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth (nth (length tail - 1) tail 0) dsu
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
nth (nth (length tail - 1) tail 0) dsu (Ancestor Unit) =
Ancestor h exact i. } }
dsu: list Slot
tail: list nat
h1: noIllegalIndices dsu
vertex: nat
h2: validChainToAncestor dsu (vertex :: tail)
IH: validChainToAncestor dsu tail
→ ∀ _0 : nat,
    nth 0 tail 0 = _0
    → ∀ _1 : nat,
        length tail ≤ _1
        → tail = ancestorChain dsu _1 _0
fuel: nat
g: ∀ _0 : nat,
  S _0 < length (vertex :: tail)
  → nth (nth _0 (vertex :: tail) 0) dsu
      (Ancestor Unit) =
    ReferTo (nth (S _0) (vertex :: tail) 0)
f: nth 0 (vertex :: tail) 0 < length dsu
e: vertex :: tail ≠ []
h4: length (vertex :: tail) ≤ S fuel
h: Tree
i: nth
  (nth (length (vertex :: tail) - 1)
     (vertex :: tail) 0) dsu 
  (Ancestor Unit) = Ancestor h
hs: length tail ≠ 0
step: nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
step2: validChainToAncestor dsu tail
→ tail = ancestorChain dsu fuel (nth 0 tail 0)
step3: validChainToAncestor dsu tail
vertex :: tail =
vertex :: ancestorChain dsu fuel (nth 0 tail 0)
    now rewrite <- (step2 step3).
Qed.

Lemma nthZeroAncestorChain (dsu : list Slot) fuel vertex : nth 0 (ancestorChain dsu fuel vertex) 0 = vertex.
dsu: list Slot
fuel, vertex: nat
nth 0 (ancestorChain dsu fuel vertex) 0 = vertex
Proof.
dsu: list Slot
fuel, vertex: nat
nth 0 (ancestorChain dsu fuel vertex) 0 = vertex
  destruct fuel as [| fuel].
dsu: list Slot
vertex: nat
nth 0 (ancestorChain dsu 0 vertex) 0 = vertex
dsu: list Slot
fuel, vertex: nat
nth 0 (ancestorChain dsu (S fuel) vertex) 0 = vertex
  -
dsu: list Slot
vertex: nat
nth 0 (ancestorChain dsu 0 vertex) 0 = vertex simpl.
dsu: list Slot
vertex: nat
vertex = vertex reflexivity.
  -
dsu: list Slot
fuel, vertex: nat
nth 0 (ancestorChain dsu (S fuel) vertex) 0 = vertex rewrite (ltac:(intros; simpl; reflexivity) : forall fuel, ancestorChain dsu (S fuel) vertex = _).
dsu: list Slot
fuel, vertex: nat
nth 0
  match nth vertex dsu (Ancestor Unit) with
  | ReferTo _0 => vertex :: ancestorChain dsu fuel _0
  | Ancestor _ => [vertex]
  end 0 = vertex destruct (nth vertex dsu (Ancestor Unit)); easy.
Qed.

Lemma validChainAncestorChain (dsu : list Slot) fuel vertex (hVertex : vertex < length dsu) (hDsu : noIllegalIndices dsu) : validChain dsu (ancestorChain dsu fuel vertex).
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
validChain dsu (ancestorChain dsu fuel vertex)
Proof.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
validChain dsu (ancestorChain dsu fuel vertex)
  induction fuel as [| fuel IH] in hVertex, vertex |- *.
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
validChain dsu (ancestorChain dsu 0 vertex)
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
validChain dsu (ancestorChain dsu (S fuel) vertex)
  {
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
validChain dsu (ancestorChain dsu 0 vertex) simpl.
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
validChain dsu [vertex] repeat split.
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
[vertex] ≠ []
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
nth 0 [vertex] 0 < length dsu
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
∀ _0 : nat,
  S _0 < length [vertex]
  → nth (nth _0 [vertex] 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) [vertex] 0) {
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
[vertex] ≠ [] easy. }
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
nth 0 [vertex] 0 < length dsu
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
∀ _0 : nat,
  S _0 < length [vertex]
  → nth (nth _0 [vertex] 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) [vertex] 0) {
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
nth 0 [vertex] 0 < length dsu simpl.
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
vertex < length dsu lia. }
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
∀ _0 : nat,
  S _0 < length [vertex]
  → nth (nth _0 [vertex] 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) [vertex] 0) {
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
∀ _0 : nat,
  S _0 < length [vertex]
  → nth (nth _0 [vertex] 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) [vertex] 0) intros x h.
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
x: nat
h: S x < length [vertex]
nth (nth x [vertex] 0) dsu (Ancestor Unit) =
ReferTo (nth (S x) [vertex] 0) simpl in h.
dsu: list Slot
vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
x: nat
h: S x < 1
nth (nth x [vertex] 0) dsu (Ancestor Unit) =
ReferTo (nth (S x) [vertex] 0) lia. } }
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
validChain dsu (ancestorChain dsu (S fuel) vertex)
  repeat split.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
ancestorChain dsu (S fuel) vertex ≠ []
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
nth 0 (ancestorChain dsu (S fuel) vertex) 0 <
length dsu
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
∀ _0 : nat,
  S _0 < length (ancestorChain dsu (S fuel) vertex)
  → nth (nth _0 (ancestorChain dsu (S fuel) vertex) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu (S fuel) vertex)
         0)
  -
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
ancestorChain dsu (S fuel) vertex ≠ [] simpl.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
match nth vertex dsu (Ancestor Unit) with
| ReferTo _0 => vertex :: ancestorChain dsu fuel _0
| Ancestor _ => [vertex]
end ≠ [] destruct (nth vertex dsu (Ancestor Unit)); easy.
  -
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
nth 0 (ancestorChain dsu (S fuel) vertex) 0 <
length dsu rewrite nthZeroAncestorChain.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
vertex < length dsu exact hVertex.
  -
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
∀ _0 : nat,
  S _0 < length (ancestorChain dsu (S fuel) vertex)
  → nth (nth _0 (ancestorChain dsu (S fuel) vertex) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu (S fuel) vertex)
         0) intros x h.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x < length (ancestorChain dsu (S fuel) vertex)
nth (nth x (ancestorChain dsu (S fuel) vertex) 0) dsu
  (Ancestor Unit) =
ReferTo
  (nth (S x) (ancestorChain dsu (S fuel) vertex) 0) simpl.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x < length (ancestorChain dsu (S fuel) vertex)
nth
  (nth x
     match nth vertex dsu (Ancestor Unit) with
     | ReferTo _0 =>
         vertex :: ancestorChain dsu fuel _0
     | Ancestor _ => [vertex]
     end 0) dsu (Ancestor Unit) =
ReferTo
  (nth (S x)
     match nth vertex dsu (Ancestor Unit) with
     | ReferTo _0 =>
         vertex :: ancestorChain dsu fuel _0
     | Ancestor _ => [vertex]
     end 0) remember (nth vertex dsu (Ancestor Unit)) as e eqn:he.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x < length (ancestorChain dsu (S fuel) vertex)
e: Slot
he: e = nth vertex dsu (Ancestor Unit)
nth
  (nth x
     match e with
     | ReferTo _0 =>
         vertex :: ancestorChain dsu fuel _0
     | Ancestor _ => [vertex]
     end 0) dsu (Ancestor Unit) =
ReferTo
  (nth (S x)
     match e with
     | ReferTo _0 =>
         vertex :: ancestorChain dsu fuel _0
     | Ancestor _ => [vertex]
     end 0) symmetry in he.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x < length (ancestorChain dsu (S fuel) vertex)
e: Slot
he: nth vertex dsu (Ancestor Unit) = e
nth
  (nth x
     match e with
     | ReferTo _0 =>
         vertex :: ancestorChain dsu fuel _0
     | Ancestor _ => [vertex]
     end 0) dsu (Ancestor Unit) =
ReferTo
  (nth (S x)
     match e with
     | ReferTo _0 =>
         vertex :: ancestorChain dsu fuel _0
     | Ancestor _ => [vertex]
     end 0) destruct e as [e | e].
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x < length (ancestorChain dsu (S fuel) vertex)
e: nat
he: nth vertex dsu (Ancestor Unit) = ReferTo e
nth (nth x (vertex :: ancestorChain dsu fuel e) 0) dsu
  (Ancestor Unit) =
ReferTo
  (nth (S x) (vertex :: ancestorChain dsu fuel e) 0)
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x < length (ancestorChain dsu (S fuel) vertex)
e: Tree
he: nth vertex dsu (Ancestor Unit) = Ancestor e
nth (nth x [vertex] 0) dsu (Ancestor Unit) =
ReferTo (nth (S x) [vertex] 0)
    +
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x < length (ancestorChain dsu (S fuel) vertex)
e: nat
he: nth vertex dsu (Ancestor Unit) = ReferTo e
nth (nth x (vertex :: ancestorChain dsu fuel e) 0) dsu
  (Ancestor Unit) =
ReferTo
  (nth (S x) (vertex :: ancestorChain dsu fuel e) 0) simpl.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x < length (ancestorChain dsu (S fuel) vertex)
e: nat
he: nth vertex dsu (Ancestor Unit) = ReferTo e
nth
  match x with
  | 0 => vertex
  | S _0 => nth _0 (ancestorChain dsu fuel e) 0
  end dsu (Ancestor Unit) =
ReferTo (nth x (ancestorChain dsu fuel e) 0) destruct x as [| x].
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
h: 1 < length (ancestorChain dsu (S fuel) vertex)
e: nat
he: nth vertex dsu (Ancestor Unit) = ReferTo e
nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 (ancestorChain dsu fuel e) 0)
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S (S x) <
length (ancestorChain dsu (S fuel) vertex)
e: nat
he: nth vertex dsu (Ancestor Unit) = ReferTo e
nth (nth x (ancestorChain dsu fuel e) 0) dsu
  (Ancestor Unit) =
ReferTo (nth (S x) (ancestorChain dsu fuel e) 0) {
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
h: 1 < length (ancestorChain dsu (S fuel) vertex)
e: nat
he: nth vertex dsu (Ancestor Unit) = ReferTo e
nth vertex dsu (Ancestor Unit) =
ReferTo (nth 0 (ancestorChain dsu fuel e) 0) rewrite nthZeroAncestorChain.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
h: 1 < length (ancestorChain dsu (S fuel) vertex)
e: nat
he: nth vertex dsu (Ancestor Unit) = ReferTo e
nth vertex dsu (Ancestor Unit) = ReferTo e exact he. }
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S (S x) <
length (ancestorChain dsu (S fuel) vertex)
e: nat
he: nth vertex dsu (Ancestor Unit) = ReferTo e
nth (nth x (ancestorChain dsu fuel e) 0) dsu
  (Ancestor Unit) =
ReferTo (nth (S x) (ancestorChain dsu fuel e) 0)
      pose proof hDsu _ _ he as hf.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S (S x) <
length (ancestorChain dsu (S fuel) vertex)
e: nat
he: nth vertex dsu (Ancestor Unit) = ReferTo e
hf: e < length dsu
nth (nth x (ancestorChain dsu fuel e) 0) dsu
  (Ancestor Unit) =
ReferTo (nth (S x) (ancestorChain dsu fuel e) 0) pose proof IH _ hf as [g1 [g2 g3]].
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S (S x) <
length (ancestorChain dsu (S fuel) vertex)
e: nat
he: nth vertex dsu (Ancestor Unit) = ReferTo e
hf: e < length dsu
g1: ancestorChain dsu fuel e ≠ []
g2: nth 0 (ancestorChain dsu fuel e) 0 < length dsu
g3: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu fuel e)
  → nth (nth _0 (ancestorChain dsu fuel e) 0) dsu
      (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu fuel e) 0)
nth (nth x (ancestorChain dsu fuel e) 0) dsu
  (Ancestor Unit) =
ReferTo (nth (S x) (ancestorChain dsu fuel e) 0)
      simpl in h.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S (S x) <
length
  match nth vertex dsu (Ancestor Unit) with
  | ReferTo _0 =>
      vertex :: ancestorChain dsu fuel _0
  | Ancestor _ => [vertex]
  end
e: nat
he: nth vertex dsu (Ancestor Unit) = ReferTo e
hf: e < length dsu
g1: ancestorChain dsu fuel e ≠ []
g2: nth 0 (ancestorChain dsu fuel e) 0 < length dsu
g3: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu fuel e)
  → nth (nth _0 (ancestorChain dsu fuel e) 0) dsu
      (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu fuel e) 0)
nth (nth x (ancestorChain dsu fuel e) 0) dsu
  (Ancestor Unit) =
ReferTo (nth (S x) (ancestorChain dsu fuel e) 0) rewrite he in h.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x, e: nat
h: S (S x) <
length (vertex :: ancestorChain dsu fuel e)
he: nth vertex dsu (Ancestor Unit) = ReferTo e
hf: e < length dsu
g1: ancestorChain dsu fuel e ≠ []
g2: nth 0 (ancestorChain dsu fuel e) 0 < length dsu
g3: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu fuel e)
  → nth (nth _0 (ancestorChain dsu fuel e) 0) dsu
      (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu fuel e) 0)
nth (nth x (ancestorChain dsu fuel e) 0) dsu
  (Ancestor Unit) =
ReferTo (nth (S x) (ancestorChain dsu fuel e) 0) simpl in h.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x, e: nat
h: S (S x) < S (length (ancestorChain dsu fuel e))
he: nth vertex dsu (Ancestor Unit) = ReferTo e
hf: e < length dsu
g1: ancestorChain dsu fuel e ≠ []
g2: nth 0 (ancestorChain dsu fuel e) 0 < length dsu
g3: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu fuel e)
  → nth (nth _0 (ancestorChain dsu fuel e) 0) dsu
      (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu fuel e) 0)
nth (nth x (ancestorChain dsu fuel e) 0) dsu
  (Ancestor Unit) =
ReferTo (nth (S x) (ancestorChain dsu fuel e) 0) exact (g3 x ltac:(lia)).
    +
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x < length (ancestorChain dsu (S fuel) vertex)
e: Tree
he: nth vertex dsu (Ancestor Unit) = Ancestor e
nth (nth x [vertex] 0) dsu (Ancestor Unit) =
ReferTo (nth (S x) [vertex] 0) simpl in h.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x <
length
  match nth vertex dsu (Ancestor Unit) with
  | ReferTo _0 =>
      vertex :: ancestorChain dsu fuel _0
  | Ancestor _ => [vertex]
  end
e: Tree
he: nth vertex dsu (Ancestor Unit) = Ancestor e
nth (nth x [vertex] 0) dsu (Ancestor Unit) =
ReferTo (nth (S x) [vertex] 0) rewrite he in h.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x < length [vertex]
e: Tree
he: nth vertex dsu (Ancestor Unit) = Ancestor e
nth (nth x [vertex] 0) dsu (Ancestor Unit) =
ReferTo (nth (S x) [vertex] 0) simpl in h.
dsu: list Slot
fuel, vertex: nat
hVertex: vertex < length dsu
hDsu: noIllegalIndices dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → validChain dsu (ancestorChain dsu fuel _0)
x: nat
h: S x < 1
e: Tree
he: nth vertex dsu (Ancestor Unit) = Ancestor e
nth (nth x [vertex] 0) dsu (Ancestor Unit) =
ReferTo (nth (S x) [vertex] 0) lia.
Qed.

Lemma validChainAncestorLength (dsu : list Slot) chain (h1 : noIllegalIndices dsu) (h2 : validChainToAncestor dsu chain) vertex (h3 : nth 0 chain 0 = vertex) : chain = ancestorChain dsu (length dsu) vertex.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
vertex: nat
h3: nth 0 chain 0 = vertex
chain = ancestorChain dsu (length dsu) vertex
Proof.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
vertex: nat
h3: nth 0 chain 0 = vertex
chain = ancestorChain dsu (length dsu) vertex apply validChainLe; try assumption.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
vertex: nat
h3: nth 0 chain 0 = vertex
length chain ≤ length dsu apply validChainMaxLength; assumption. Qed.

Lemma validChainTake (dsu : list Slot) chain (h1 : noIllegalIndices dsu) (h2 : validChainToAncestor dsu chain) fuel vertex (h3 : nth 0 chain 0 = vertex) : take (S fuel) chain = ancestorChain dsu fuel vertex.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
take (S fuel) chain = ancestorChain dsu fuel vertex
Proof.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
take (S fuel) chain = ancestorChain dsu fuel vertex
  induction fuel as [| fuel IH].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
vertex: nat
h3: nth 0 chain 0 = vertex
take 1 chain = ancestorChain dsu 0 vertex
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
take (S (S fuel)) chain =
ancestorChain dsu (S fuel) vertex
  {
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
vertex: nat
h3: nth 0 chain 0 = vertex
take 1 chain = ancestorChain dsu 0 vertex destruct chain as [| head tail].
dsu: list Slot
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu []
vertex: nat
h3: nth 0 [] 0 = vertex
take 1 [] = ancestorChain dsu 0 vertex
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
take 1 (head :: tail) = ancestorChain dsu 0 vertex
    -
dsu: list Slot
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu []
vertex: nat
h3: nth 0 [] 0 = vertex
take 1 [] = ancestorChain dsu 0 vertex destruct h2 as [[e [f g]] [b c]].
dsu: list Slot
h1: noIllegalIndices dsu
e: [] ≠ []
f: nth 0 [] 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length []
  → nth (nth _0 [] 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) [] 0)
b: Tree
c: nth (nth (length [] - 1) [] 0) dsu (Ancestor Unit) =
Ancestor b
vertex: nat
h3: nth 0 [] 0 = vertex
take 1 [] = ancestorChain dsu 0 vertex exfalso.
dsu: list Slot
h1: noIllegalIndices dsu
e: [] ≠ []
f: nth 0 [] 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length []
  → nth (nth _0 [] 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) [] 0)
b: Tree
c: nth (nth (length [] - 1) [] 0) dsu (Ancestor Unit) =
Ancestor b
vertex: nat
h3: nth 0 [] 0 = vertex
False exact (e ltac:(reflexivity)).
    -
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
vertex: nat
h3: nth 0 (head :: tail) 0 = vertex
take 1 (head :: tail) = ancestorChain dsu 0 vertex simpl in *.
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
vertex: nat
h3: head = vertex
head :: take 0 tail = [vertex] rewrite take_0, h3.
dsu: list Slot
head: nat
tail: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu (head :: tail)
vertex: nat
h3: head = vertex
[vertex] = [vertex] reflexivity. }
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
take (S (S fuel)) chain =
ancestorChain dsu (S fuel) vertex
  pose proof take_S_r chain (S fuel) (nth (S fuel) chain 0) as h4.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
take (S (S fuel)) chain =
ancestorChain dsu (S fuel) vertex
  pose proof nth_lookup_or_length chain (S fuel) 0 as [h5 | h5].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
take (S (S fuel)) chain =
ancestorChain dsu (S fuel) vertex
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: length chain ≤ S fuel
take (S (S fuel)) chain =
ancestorChain dsu (S fuel) vertex
  -
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
take (S (S fuel)) chain =
ancestorChain dsu (S fuel) vertex rewrite (h4 h5), IH.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
ancestorChain dsu fuel vertex ++
[nth (S fuel) chain 0] =
ancestorChain dsu (S fuel) vertex pose proof lookup_lt_Some _ _ _ h5 as h6.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
h6: S fuel < length chain
ancestorChain dsu fuel vertex ++
[nth (S fuel) chain 0] =
ancestorChain dsu (S fuel) vertex
    assert (h7 : forall delta, S fuel + delta < length chain -> ancestorChain dsu fuel (nth delta chain 0) ++ [nth (S fuel + delta) chain 0] = ancestorChain dsu (S fuel) (nth delta chain 0)).
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
h6: S fuel < length chain
∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
h6: S fuel < length chain
h7: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
ancestorChain dsu fuel vertex ++
[nth (S fuel) chain 0] =
ancestorChain dsu (S fuel) vertex
    {
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
h6: S fuel < length chain
∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0) revert h2.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
h6: S fuel < length chain
validChainToAncestor dsu chain
→ ∀ _0 : nat,
    S fuel + _0 < length chain
    → ancestorChain dsu fuel (nth _0 chain 0) ++
      [nth (S fuel + _0) chain 0] =
      ancestorChain dsu (S fuel) (nth _0 chain 0) clear.
dsu: list Slot
chain: list nat
fuel: nat
validChainToAncestor dsu chain
→ ∀ _0 : nat,
    S fuel + _0 < length chain
    → ancestorChain dsu fuel (nth _0 chain 0) ++
      [nth (S fuel + _0) chain 0] =
      ancestorChain dsu (S fuel) (nth _0 chain 0) intro h2.
dsu: list Slot
chain: list nat
fuel: nat
h2: validChainToAncestor dsu chain
∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0) induction fuel as [| fuel IH]; intros delta h7.
dsu: list Slot
chain: list nat
h2: validChainToAncestor dsu chain
delta: nat
h7: 1 + delta < length chain
ancestorChain dsu 0 (nth delta chain 0) ++
[nth (1 + delta) chain 0] =
ancestorChain dsu 1 (nth delta chain 0)
dsu: list Slot
chain: list nat
fuel: nat
h2: validChainToAncestor dsu chain
IH: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
delta: nat
h7: S (S fuel) + delta < length chain
ancestorChain dsu (S fuel) (nth delta chain 0) ++
[nth (S (S fuel) + delta) chain 0] =
ancestorChain dsu (S (S fuel)) (nth delta chain 0)
      -
dsu: list Slot
chain: list nat
h2: validChainToAncestor dsu chain
delta: nat
h7: 1 + delta < length chain
ancestorChain dsu 0 (nth delta chain 0) ++
[nth (1 + delta) chain 0] =
ancestorChain dsu 1 (nth delta chain 0) simpl.
dsu: list Slot
chain: list nat
h2: validChainToAncestor dsu chain
delta: nat
h7: 1 + delta < length chain
[nth delta chain 0; nth (S delta) chain 0] =
match nth (nth delta chain 0) dsu (Ancestor Unit) with
| ReferTo _0 => [nth delta chain 0; _0]
| Ancestor _ => [nth delta chain 0]
end pose proof h2 as [[e [f g]] [b c]].
dsu: list Slot
chain: list nat
h2: validChainToAncestor dsu chain
delta: nat
h7: 1 + delta < length chain
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
[nth delta chain 0; nth (S delta) chain 0] =
match nth (nth delta chain 0) dsu (Ancestor Unit) with
| ReferTo _0 => [nth delta chain 0; _0]
| Ancestor _ => [nth delta chain 0]
end rewrite g; [| lia].
dsu: list Slot
chain: list nat
h2: validChainToAncestor dsu chain
delta: nat
h7: 1 + delta < length chain
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
[nth delta chain 0; nth (S delta) chain 0] =
[nth delta chain 0; nth (S delta) chain 0] reflexivity.
      -
dsu: list Slot
chain: list nat
fuel: nat
h2: validChainToAncestor dsu chain
IH: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
delta: nat
h7: S (S fuel) + delta < length chain
ancestorChain dsu (S fuel) (nth delta chain 0) ++
[nth (S (S fuel) + delta) chain 0] =
ancestorChain dsu (S (S fuel)) (nth delta chain 0) rewrite (ltac:(simpl; reflexivity) : ancestorChain _ (S _) _ = _).
dsu: list Slot
chain: list nat
fuel: nat
h2: validChainToAncestor dsu chain
IH: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
delta: nat
h7: S (S fuel) + delta < length chain
match nth (nth delta chain 0) dsu (Ancestor Unit) with
| ReferTo _0 =>
    nth delta chain 0 :: ancestorChain dsu fuel _0
| Ancestor _ => [nth delta chain 0]
end ++ [nth (S (S fuel) + delta) chain 0] =
ancestorChain dsu (S (S fuel)) (nth delta chain 0) pose proof h2 as [[e [f g]] [b c]].
dsu: list Slot
chain: list nat
fuel: nat
h2: validChainToAncestor dsu chain
IH: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
delta: nat
h7: S (S fuel) + delta < length chain
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
match nth (nth delta chain 0) dsu (Ancestor Unit) with
| ReferTo _0 =>
    nth delta chain 0 :: ancestorChain dsu fuel _0
| Ancestor _ => [nth delta chain 0]
end ++ [nth (S (S fuel) + delta) chain 0] =
ancestorChain dsu (S (S fuel)) (nth delta chain 0) rewrite g; [| lia].
dsu: list Slot
chain: list nat
fuel: nat
h2: validChainToAncestor dsu chain
IH: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
delta: nat
h7: S (S fuel) + delta < length chain
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
(nth delta chain 0
 :: ancestorChain dsu fuel (nth (S delta) chain 0)) ++
[nth (S (S fuel) + delta) chain 0] =
ancestorChain dsu (S (S fuel)) (nth delta chain 0) rewrite (ltac:(intros; listsEqual) : forall a b c, (a :: b) ++ [c] = a :: (b ++ [c])), (ltac:(lia) : S (S fuel) + delta = S fuel + S delta), IH; [| lia].
dsu: list Slot
chain: list nat
fuel: nat
h2: validChainToAncestor dsu chain
IH: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
delta: nat
h7: S (S fuel) + delta < length chain
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
nth delta chain 0
:: ancestorChain dsu (S fuel) (nth (S delta) chain 0) =
ancestorChain dsu (S (S fuel)) (nth delta chain 0) rewrite (ltac:(intro x; simpl; reflexivity) : forall x, ancestorChain _ (S x) (nth delta chain 0) = _).
dsu: list Slot
chain: list nat
fuel: nat
h2: validChainToAncestor dsu chain
IH: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
delta: nat
h7: S (S fuel) + delta < length chain
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
nth delta chain 0
:: ancestorChain dsu (S fuel) (nth (S delta) chain 0) =
match nth (nth delta chain 0) dsu (Ancestor Unit) with
| ReferTo _0 =>
    nth delta chain 0 :: ancestorChain dsu (S fuel) _0
| Ancestor _ => [nth delta chain 0]
end rewrite g; [| lia].
dsu: list Slot
chain: list nat
fuel: nat
h2: validChainToAncestor dsu chain
IH: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
delta: nat
h7: S (S fuel) + delta < length chain
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
nth delta chain 0
:: ancestorChain dsu (S fuel) (nth (S delta) chain 0) =
nth delta chain 0
:: ancestorChain dsu (S fuel) (nth (S delta) chain 0) reflexivity. }
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
h6: S fuel < length chain
h7: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
ancestorChain dsu fuel vertex ++
[nth (S fuel) chain 0] =
ancestorChain dsu (S fuel) vertex
    pose proof h7 0 ltac:(lia).
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
h6: S fuel < length chain
h7: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
_0: ancestorChain dsu fuel (nth 0 chain 0) ++
[nth (S fuel + 0) chain 0] =
ancestorChain dsu (S fuel) (nth 0 chain 0)
ancestorChain dsu fuel vertex ++
[nth (S fuel) chain 0] =
ancestorChain dsu (S fuel) vertex rewrite (ltac:(easy) : forall x, x + 0 = x) in *.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
h6: S fuel < length chain
h7: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
_0: ancestorChain dsu fuel (nth 0 chain 0) ++
[nth (S fuel) chain 0] =
ancestorChain dsu (S fuel) (nth 0 chain 0)
ancestorChain dsu fuel vertex ++
[nth (S fuel) chain 0] =
ancestorChain dsu (S fuel) vertex subst vertex.
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel: nat
IH: take (S fuel) chain =
ancestorChain dsu fuel (nth 0 chain 0)
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: chain !! S fuel = Some (nth (S fuel) chain 0)
h6: S fuel < length chain
h7: ∀ _0 : nat,
  S fuel + _0 < length chain
  → ancestorChain dsu fuel (nth _0 chain 0) ++
    [nth (S fuel + _0) chain 0] =
    ancestorChain dsu (S fuel) (nth _0 chain 0)
_0: ancestorChain dsu fuel (nth 0 chain 0) ++
[nth (S fuel) chain 0] =
ancestorChain dsu (S fuel) (nth 0 chain 0)
ancestorChain dsu fuel (nth 0 chain 0) ++
[nth (S fuel) chain 0] =
ancestorChain dsu (S fuel) (nth 0 chain 0) assumption.
  -
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: length chain ≤ S fuel
take (S (S fuel)) chain =
ancestorChain dsu (S fuel) vertex rewrite firstn_all2; [| lia].
dsu: list Slot
chain: list nat
h1: noIllegalIndices dsu
h2: validChainToAncestor dsu chain
fuel, vertex: nat
h3: nth 0 chain 0 = vertex
IH: take (S fuel) chain =
ancestorChain dsu fuel vertex
h4: chain !! S fuel = Some (nth (S fuel) chain 0)
→ take (S (S fuel)) chain =
  take (S fuel) chain ++ [nth (S fuel) chain 0]
h5: length chain ≤ S fuel
chain = ancestorChain dsu (S fuel) vertex
    exact (validChainLe dsu chain h1 h2 (S fuel) vertex h3 h5).
Qed.

Lemma ancestorLtLength dsu (h : noIllegalIndices dsu) n index (h1 : index < length dsu) : ancestor dsu n index < length dsu.
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
ancestor dsu n index < length dsu
Proof.
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
ancestor dsu n index < length dsu
  induction n as [| n IH] in h1, index |- *; [easy |].
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → ancestor dsu n _0 < length dsu
ancestor dsu (S n) index < length dsu
  simpl.
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → ancestor dsu n _0 < length dsu
match nth index dsu (Ancestor Unit) with
| ReferTo _0 => ancestor dsu n _0
| Ancestor _ => index
end < length dsu remember (nth index dsu (Ancestor Unit)) as x eqn:hX.
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → ancestor dsu n _0 < length dsu
x: Slot
hX: x = nth index dsu (Ancestor Unit)
match x with
| ReferTo _0 => ancestor dsu n _0
| Ancestor _ => index
end < length dsu symmetry in hX.
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → ancestor dsu n _0 < length dsu
x: Slot
hX: nth index dsu (Ancestor Unit) = x
match x with
| ReferTo _0 => ancestor dsu n _0
| Ancestor _ => index
end < length dsu destruct x as [x | x].
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → ancestor dsu n _0 < length dsu
x: nat
hX: nth index dsu (Ancestor Unit) = ReferTo x
ancestor dsu n x < length dsu
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → ancestor dsu n _0 < length dsu
x: Tree
hX: nth index dsu (Ancestor Unit) = Ancestor x
index < length dsu
  -
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → ancestor dsu n _0 < length dsu
x: nat
hX: nth index dsu (Ancestor Unit) = ReferTo x
ancestor dsu n x < length dsu pose proof h _ _ hX.
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → ancestor dsu n _0 < length dsu
x: nat
hX: nth index dsu (Ancestor Unit) = ReferTo x
_0: x < length dsu
ancestor dsu n x < length dsu apply IH.
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → ancestor dsu n _0 < length dsu
x: nat
hX: nth index dsu (Ancestor Unit) = ReferTo x
_0: x < length dsu
x < length dsu lia.
  -
dsu: list Slot
h: noIllegalIndices dsu
n, index: nat
h1: index < length dsu
IH: ∀ _0 : nat,
  _0 < length dsu
  → ancestor dsu n _0 < length dsu
x: Tree
hX: nth index dsu (Ancestor Unit) = Ancestor x
index < length dsu exact h1.
Qed.

Definition withoutCyclesN (dsu : list Slot) (n : nat) :=
  forall x, x < n -> match nth (ancestor dsu (length dsu) x) dsu (Ancestor Unit) with
  | Ancestor _ => true
  | _ => false
  end.

Lemma ancestorReferTo (dsu : list Slot) (u v : nat) (hU : u < length dsu) (h : noIllegalIndices dsu) (h' : withoutCyclesN dsu (length dsu)) (hR : nth u dsu (Ancestor Unit) = ReferTo v) : ancestor dsu (length dsu) u = ancestor dsu (length dsu) v.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
ancestor dsu (length dsu) u =
ancestor dsu (length dsu) v
Proof.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
ancestor dsu (length dsu) u =
ancestor dsu (length dsu) v
  rewrite <- !ancestorEqLastAncestorChain.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
nth (length (ancestorChain dsu (length dsu) u) - 1)
  (ancestorChain dsu (length dsu) u) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0
  pose proof h u v hR as hV.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
nth (length (ancestorChain dsu (length dsu) u) - 1)
  (ancestorChain dsu (length dsu) u) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0
  assert (hL : 0 < length dsu).
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
0 < length dsu
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
nth (length (ancestorChain dsu (length dsu) u) - 1)
  (ancestorChain dsu (length dsu) u) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0
  {
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
0 < length dsu destruct dsu as [| head tail].
u, v: nat
hU: u < length []
h: noIllegalIndices []
h': withoutCyclesN [] (length [])
hR: nth u [] (Ancestor Unit) = ReferTo v
hV: v < length []
0 < length []
head: Slot
tail: list Slot
u, v: nat
hU: u < length (head :: tail)
h: noIllegalIndices (head :: tail)
h': withoutCyclesN (head :: tail)
  (length (head :: tail))
hR: nth u (head :: tail) (Ancestor Unit) = ReferTo v
hV: v < length (head :: tail)
0 < length (head :: tail) {
u, v: nat
hU: u < length []
h: noIllegalIndices []
h': withoutCyclesN [] (length [])
hR: nth u [] (Ancestor Unit) = ReferTo v
hV: v < length []
0 < length [] simpl in hR.
u, v: nat
hU: u < length []
h: noIllegalIndices []
h': withoutCyclesN [] (length [])
hR: match u with
| 0 | _ => Ancestor Unit
end = ReferTo v
hV: v < length []
0 < length [] easy. }
head: Slot
tail: list Slot
u, v: nat
hU: u < length (head :: tail)
h: noIllegalIndices (head :: tail)
h': withoutCyclesN (head :: tail)
  (length (head :: tail))
hR: nth u (head :: tail) (Ancestor Unit) = ReferTo v
hV: v < length (head :: tail)
0 < length (head :: tail)
    simpl.
head: Slot
tail: list Slot
u, v: nat
hU: u < length (head :: tail)
h: noIllegalIndices (head :: tail)
h': withoutCyclesN (head :: tail)
  (length (head :: tail))
hR: nth u (head :: tail) (Ancestor Unit) = ReferTo v
hV: v < length (head :: tail)
0 < S (length tail) lia.  }
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
nth (length (ancestorChain dsu (length dsu) u) - 1)
  (ancestorChain dsu (length dsu) u) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0
  rewrite (ltac:(lia) : length dsu = S (length dsu - 1)) at 1.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
nth
  (length (ancestorChain dsu (S (length dsu - 1)) u) -
   1) (ancestorChain dsu (length dsu) u) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 simpl.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
nth
  (length
     match nth u dsu (Ancestor Unit) with
     | ReferTo _0 =>
         u :: ancestorChain dsu (length dsu - 1) _0
     | Ancestor _ => [u]
     end - 1) (ancestorChain dsu (length dsu) u) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 rewrite hR.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
nth
  (length (u :: ancestorChain dsu (length dsu - 1) v) -
   1) (ancestorChain dsu (length dsu) u) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 rewrite (ltac:(lia) : length dsu = S (length dsu - 1)) at 2.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
nth
  (length (u :: ancestorChain dsu (length dsu - 1) v) -
   1) (ancestorChain dsu (S (length dsu - 1)) u) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 simpl.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
nth
  (length (ancestorChain dsu (length dsu - 1) v) - 0)
  match nth u dsu (Ancestor Unit) with
  | ReferTo _0 =>
      u :: ancestorChain dsu (length dsu - 1) _0
  | Ancestor _ => [u]
  end 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 rewrite hR, Nat.sub_0_r.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
nth (length (ancestorChain dsu (length dsu - 1) v))
  (u :: ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0
  pose proof h' u hU as hB.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
hB: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
nth (length (ancestorChain dsu (length dsu - 1) v))
  (u :: ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 rewrite <- ancestorEqLastAncestorChain in hB.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
hB: match
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) -
        1) (ancestorChain dsu (length dsu) u) 0)
    dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
nth (length (ancestorChain dsu (length dsu - 1) v))
  (u :: ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 rewrite (ltac:(lia) : length dsu = S (length dsu - 1)) in hB.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
hB: match
  nth
    (nth
       (length
          (ancestorChain dsu 
             (S (length dsu - 1)) u) - 1)
       (ancestorChain dsu (S (length dsu - 1)) u)
       0) dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
nth (length (ancestorChain dsu (length dsu - 1) v))
  (u :: ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 simpl in hB.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
hB: match
  nth
    (nth
       (length
          match nth u dsu (Ancestor Unit) with
          | ReferTo _0 =>
              u
              :: ancestorChain dsu
                   (length dsu - 1) _0
          | Ancestor _ => [u]
          end - 1)
       match nth u dsu (Ancestor Unit) with
       | ReferTo _0 =>
           u
           :: ancestorChain dsu 
                (length dsu - 1) _0
       | Ancestor _ => [u]
       end 0) dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
nth (length (ancestorChain dsu (length dsu - 1) v))
  (u :: ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 rewrite hR in hB.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
hB: match
  nth
    (nth
       (length
          (u
           :: ancestorChain dsu 
                (length dsu - 1) v) - 1)
       (u :: ancestorChain dsu (length dsu - 1) v)
       0) dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
nth (length (ancestorChain dsu (length dsu - 1) v))
  (u :: ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 simpl in hB.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
hB: match
  nth
    match
      length
        (ancestorChain dsu (length dsu - 1) v) -
      0
    with
    | 0 => u
    | S _0 =>
        nth _0
          (ancestorChain dsu (length dsu - 1) v)
          0
    end dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
nth (length (ancestorChain dsu (length dsu - 1) v))
  (u :: ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 rewrite Nat.sub_0_r in hB.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
hB: match
  nth
    match
      length
        (ancestorChain dsu (length dsu - 1) v)
    with
    | 0 => u
    | S _0 =>
        nth _0
          (ancestorChain dsu (length dsu - 1) v)
          0
    end dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
nth (length (ancestorChain dsu (length dsu - 1) v))
  (u :: ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0
  pose proof zeroLtLengthAncestorChain dsu (length dsu - 1) v as hA.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
hB: match
  nth
    match
      length
        (ancestorChain dsu (length dsu - 1) v)
    with
    | 0 => u
    | S _0 =>
        nth _0
          (ancestorChain dsu (length dsu - 1) v)
          0
    end dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth (length (ancestorChain dsu (length dsu - 1) v))
  (u :: ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0
  rewrite ((ltac:(lia) : forall a, 0 < a -> a = S (a - 1)) _ hA) in *.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
hB: match
  nth
    (nth
       (length
          (ancestorChain dsu (length dsu - 1) v) -
        1) (ancestorChain dsu (length dsu - 1) v)
       0) dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth
  (S
     (length (ancestorChain dsu (length dsu - 1) v) -
      1)) (u :: ancestorChain dsu (length dsu - 1) v)
  0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 simpl.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
hB: match
  nth
    (nth
       (length
          (ancestorChain dsu (length dsu - 1) v) -
        1) (ancestorChain dsu (length dsu - 1) v)
       0) dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth
  (length (ancestorChain dsu (length dsu - 1) v) - 1)
  (ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 remember (nth
    (nth (length (ancestorChain dsu (length dsu - 1) v) - 1)
       (ancestorChain dsu (length dsu - 1) v) 0) dsu 
    (Ancestor Unit)) as e eqn:he.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Slot
he: e =
nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit)
hB: match e with
| ReferTo _ => false
| Ancestor _ => true
end
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth
  (length (ancestorChain dsu (length dsu - 1) v) - 1)
  (ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 destruct e as [e | e].
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: nat
he: ReferTo e =
nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit)
hB: false
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth
  (length (ancestorChain dsu (length dsu - 1) v) - 1)
  (ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: Ancestor e =
nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit)
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth
  (length (ancestorChain dsu (length dsu - 1) v) - 1)
  (ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 {
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: nat
he: ReferTo e =
nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit)
hB: false
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth
  (length (ancestorChain dsu (length dsu - 1) v) - 1)
  (ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 easy. }
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: Ancestor e =
nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit)
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth
  (length (ancestorChain dsu (length dsu - 1) v) - 1)
  (ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0
  symmetry in he.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit) = Ancestor e
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth
  (length (ancestorChain dsu (length dsu - 1) v) - 1)
  (ancestorChain dsu (length dsu - 1) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0
  rewrite (fun i => validChainAncestorLength dsu (ancestorChain dsu (length dsu - 1) v) h i v ltac:(apply firstAncestorChain)).
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit) = Ancestor e
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit) = Ancestor e
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
validChainToAncestor dsu
  (ancestorChain dsu (length dsu - 1) v) {
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit) = Ancestor e
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 =
nth (length (ancestorChain dsu (length dsu) v) - 1)
  (ancestorChain dsu (length dsu) v) 0 reflexivity. }
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit) = Ancestor e
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
validChainToAncestor dsu
  (ancestorChain dsu (length dsu - 1) v)
  split.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit) = Ancestor e
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
validChain dsu (ancestorChain dsu (length dsu - 1) v)
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit) = Ancestor e
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu - 1) v) -
        1) (ancestorChain dsu (length dsu - 1) v) 0)
    dsu (Ancestor Unit) = 
  Ancestor _0
  -
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit) = Ancestor e
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
validChain dsu (ancestorChain dsu (length dsu - 1) v) apply validChainAncestorChain; assumption.
  -
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit) = Ancestor e
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu - 1) v) -
        1) (ancestorChain dsu (length dsu - 1) v) 0)
    dsu (Ancestor Unit) = 
  Ancestor _0 exists e.
dsu: list Slot
u, v: nat
hU: u < length dsu
h: noIllegalIndices dsu
h': withoutCyclesN dsu (length dsu)
hR: nth u dsu (Ancestor Unit) = ReferTo v
hV: v < length dsu
hL: 0 < length dsu
e: Tree
he: nth
  (nth
     (length
        (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0)
  dsu (Ancestor Unit) = Ancestor e
hB: true
hA: 0 < length (ancestorChain dsu (length dsu - 1) v)
nth
  (nth
     (length (ancestorChain dsu (length dsu - 1) v) -
      1) (ancestorChain dsu (length dsu - 1) v) 0) dsu
  (Ancestor Unit) = Ancestor e exact he.
Qed.

Fixpoint withoutCyclesBool (dsu : list Slot) (n : nat) : bool :=
  match n with
  | O => true
  | S n => match nth (ancestor dsu (length dsu) n) dsu (Ancestor Unit) with
    | Ancestor _ => withoutCyclesBool dsu n
    | _ => false
    end
  end.

Lemma withoutCyclesNIffWithoutCyclesBool dsu n : withoutCyclesN dsu n <-> withoutCyclesBool dsu n.
dsu: list Slot
n: nat
withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
Proof.
dsu: list Slot
n: nat
withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
  induction n as [| n IH].
dsu: list Slot
withoutCyclesN dsu 0 ↔ withoutCyclesBool dsu 0
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
withoutCyclesN dsu (S n) ↔ withoutCyclesBool dsu (S n)
  -
dsu: list Slot
withoutCyclesN dsu 0 ↔ withoutCyclesBool dsu 0 simpl.
dsu: list Slot
withoutCyclesN dsu 0 ↔ True unfold withoutCyclesN.
dsu: list Slot
(∀ _0 : nat,
   _0 < 0
   → match
       nth (ancestor dsu (length dsu) _0) dsu
         (Ancestor Unit)
     with
     | ReferTo _ => false
     | Ancestor _ => true
     end) ↔ True split; intros; lia.
  -
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
withoutCyclesN dsu (S n) ↔ withoutCyclesBool dsu (S n) split; intro h.
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: withoutCyclesN dsu (S n)
withoutCyclesBool dsu (S n)
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: withoutCyclesBool dsu (S n)
withoutCyclesN dsu (S n)
  {
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: withoutCyclesN dsu (S n)
withoutCyclesBool dsu (S n) rewrite (ltac:(easy) : withoutCyclesBool dsu (S n) = match nth (ancestor dsu (length dsu) n) dsu (Ancestor Unit) with
    | Ancestor _ => withoutCyclesBool dsu n
    | _ => false
    end).
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: withoutCyclesN dsu (S n)
match
  nth (ancestor dsu (length dsu) n) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => withoutCyclesBool dsu n
end
    pose proof h n ltac:(lia) as step.
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: withoutCyclesN dsu (S n)
step: match
  nth (ancestor dsu (length dsu) n) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
match
  nth (ancestor dsu (length dsu) n) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => withoutCyclesBool dsu n
end destruct (nth (ancestor dsu (length dsu) n) dsu (Ancestor Unit)) as [h3 |]; [contradiction; exact h3 |].
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: withoutCyclesN dsu (S n)
_0: Tree
step: true
withoutCyclesBool dsu n rewrite <- IH.
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: withoutCyclesN dsu (S n)
_0: Tree
step: true
withoutCyclesN dsu n intros x y.
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: withoutCyclesN dsu (S n)
_0: Tree
step: true
x: nat
y: x < n
match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end pose proof h x ltac:(lia).
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: withoutCyclesN dsu (S n)
_0: Tree
step: true
x: nat
y: x < n
_1: match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end assumption. }
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: withoutCyclesBool dsu (S n)
withoutCyclesN dsu (S n)
  {
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: withoutCyclesBool dsu (S n)
withoutCyclesN dsu (S n) rewrite (ltac:(easy) : withoutCyclesBool dsu (S n) = match nth (ancestor dsu (length dsu) n) dsu (Ancestor Unit) with
    | Ancestor _ => withoutCyclesBool dsu n
    | _ => false
    end) in h.
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: match
  nth (ancestor dsu (length dsu) n) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => withoutCyclesBool dsu n
end
withoutCyclesN dsu (S n) intros x y.
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: match
  nth (ancestor dsu (length dsu) n) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => withoutCyclesBool dsu n
end
x: nat
y: x < S n
match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end destruct (ltac:(lia) : x = n \/ x < n) as [h2 | h2].
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: match
  nth (ancestor dsu (length dsu) n) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => withoutCyclesBool dsu n
end
x: nat
y: x < S n
h2: x = n
match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: match
  nth (ancestor dsu (length dsu) n) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => withoutCyclesBool dsu n
end
x: nat
y: x < S n
h2: x < n
match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end {
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: match
  nth (ancestor dsu (length dsu) n) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => withoutCyclesBool dsu n
end
x: nat
y: x < S n
h2: x = n
match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite h2.
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: match
  nth (ancestor dsu (length dsu) n) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => withoutCyclesBool dsu n
end
x: nat
y: x < S n
h2: x = n
match
  nth (ancestor dsu (length dsu) n) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end destruct (nth (ancestor dsu (length dsu) n) dsu (Ancestor Unit)) as [h3 | h3]; [assumption |].
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h3: Tree
h: withoutCyclesBool dsu n
x: nat
y: x < S n
h2: x = n
true easy. }
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
h: match
  nth (ancestor dsu (length dsu) n) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => withoutCyclesBool dsu n
end
x: nat
y: x < S n
h2: x < n
match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end destruct (nth (ancestor dsu (length dsu) n) dsu (Ancestor Unit)) as [h1 |]; [contradiction; exact h1 |].
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
_0: Tree
h: withoutCyclesBool dsu n
x: nat
y: x < S n
h2: x < n
match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite <- IH in h.
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
_0: Tree
h: withoutCyclesN dsu n
x: nat
y: x < S n
h2: x < n
match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end pose proof h x ltac:(lia).
dsu: list Slot
n: nat
IH: withoutCyclesN dsu n ↔ withoutCyclesBool dsu n
_0: Tree
h: withoutCyclesN dsu n
x: nat
y: x < S n
h2: x < n
_1: match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
match
  nth (ancestor dsu (length dsu) x) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end assumption. }
Qed.

Lemma ancestorChainInsertNotPresent (dsu : list Slot) (fuel : nat) (u x : nat) (h1 : noIllegalIndices dsu) (h2 : withoutCyclesN dsu (length dsu)) (h3 : u < length dsu) (h4 : x < length dsu) (h5 : match nth x dsu (Ancestor Unit) with | ReferTo _ => true | Ancestor _ => false end) (hd : ∀ _0 : nat, _0 < length (ancestorChain dsu (length dsu) u) → nth _0 (ancestorChain dsu (length dsu) u) 0 ≠ x) : ancestorChain dsu fuel u = ancestorChain (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) fuel u.
dsu: list Slot
fuel, u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
ancestorChain dsu fuel u =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  fuel u
Proof.
dsu: list Slot
fuel, u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
ancestorChain dsu fuel u =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  fuel u
  revert h3 hd.
dsu: list Slot
fuel, u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
u < length dsu
→ (∀ _0 : nat,
     _0 < length (ancestorChain dsu (length dsu) u)
     → nth _0 (ancestorChain dsu (length dsu) u) 0 ≠ x)
  → ancestorChain dsu fuel u =
    ancestorChain
      (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
         dsu) fuel u revert u.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 < length (ancestorChain dsu (length dsu) _0)
       → nth _1 (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
           dsu) fuel _0 induction fuel as [| fuel IH]; intros u h3 hd.
dsu: list Slot
x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
ancestorChain dsu 0 u =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) 0
  u
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
ancestorChain dsu (S fuel) u =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (S fuel) u {
dsu: list Slot
x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
ancestorChain dsu 0 u =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) 0
  u easy. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
ancestorChain dsu (S fuel) u =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (S fuel) u
  simpl.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
match nth u dsu (Ancestor Unit) with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match
  nth u
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit)
with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end pose proof hd 0 ltac:(destruct (length dsu); simpl; destruct (nth u dsu (Ancestor Unit)); simpl; try lia) as step1.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: nth 0 (ancestorChain dsu (length dsu) u) 0 ≠ x
match nth u dsu (Ancestor Unit) with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match
  nth u
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit)
with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end
  assert (step2 : nth 0 (ancestorChain dsu (length dsu) u) 0 = u).
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: nth 0 (ancestorChain dsu (length dsu) u) 0 ≠ x
nth 0 (ancestorChain dsu (length dsu) u) 0 = u
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: nth 0 (ancestorChain dsu (length dsu) u) 0 ≠ x
step2: nth 0 (ancestorChain dsu (length dsu) u) 0 = u
match nth u dsu (Ancestor Unit) with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match
  nth u
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit)
with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end
  {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: nth 0 (ancestorChain dsu (length dsu) u) 0 ≠ x
nth 0 (ancestorChain dsu (length dsu) u) 0 = u destruct (length dsu); [reflexivity |].
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
_0: nat
h2: withoutCyclesN dsu (S _0)
h4: x < S _0
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _1 : nat,
  _1 < S _0
  → (∀ _2 : nat,
       _2 < length (ancestorChain dsu (S _0) _1)
       → nth _2 (ancestorChain dsu (S _0) _1) 0
         ≠ x)
    → ancestorChain dsu fuel _1 =
      ancestorChain
        (<[x:=ReferTo (ancestor dsu (S _0) x)]>
           dsu) fuel _1
u: nat
h3: u < S _0
hd: ∀ _1 : nat,
  _1 < length (ancestorChain dsu (S _0) u)
  → nth _1 (ancestorChain dsu (S _0) u) 0 ≠ x
step1: nth 0 (ancestorChain dsu (S _0) u) 0 ≠ x
nth 0 (ancestorChain dsu (S _0) u) 0 = u simpl.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
_0: nat
h2: withoutCyclesN dsu (S _0)
h4: x < S _0
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _1 : nat,
  _1 < S _0
  → (∀ _2 : nat,
       _2 < length (ancestorChain dsu (S _0) _1)
       → nth _2 (ancestorChain dsu (S _0) _1) 0
         ≠ x)
    → ancestorChain dsu fuel _1 =
      ancestorChain
        (<[x:=ReferTo (ancestor dsu (S _0) x)]>
           dsu) fuel _1
u: nat
h3: u < S _0
hd: ∀ _1 : nat,
  _1 < length (ancestorChain dsu (S _0) u)
  → nth _1 (ancestorChain dsu (S _0) u) 0 ≠ x
step1: nth 0 (ancestorChain dsu (S _0) u) 0 ≠ x
nth 0
  match nth u dsu (Ancestor Unit) with
  | ReferTo _1 => u :: ancestorChain dsu _0 _1
  | Ancestor _ => [u]
  end 0 = u destruct (nth u dsu (Ancestor Unit)); easy. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: nth 0 (ancestorChain dsu (length dsu) u) 0 ≠ x
step2: nth 0 (ancestorChain dsu (length dsu) u) 0 = u
match nth u dsu (Ancestor Unit) with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match
  nth u
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit)
with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end
  rewrite step2 in step1.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: nth 0 (ancestorChain dsu (length dsu) u) 0 = u
match nth u dsu (Ancestor Unit) with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match
  nth u
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit)
with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end clear step2.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
match nth u dsu (Ancestor Unit) with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match
  nth u
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit)
with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end pose proof list_lookup_insert_ne dsu x u (ReferTo (ancestor dsu (length dsu) x)) ltac:(intro hh; subst x; exact (step1 ltac:(reflexivity))) as step2.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
match nth u dsu (Ancestor Unit) with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match
  nth u
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit)
with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end
  pose proof (ltac:(intros a b h; subst a; reflexivity): forall a b, a = b -> default (Ancestor Unit) a = default (Ancestor Unit) b) _ _ step2 as step3.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
step3: default (Ancestor Unit)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu !! u) =
default (Ancestor Unit) (dsu !! u)
match nth u dsu (Ancestor Unit) with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match
  nth u
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit)
with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end rewrite <- !nth_lookup in step3.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) =
nth u dsu (Ancestor Unit)
match nth u dsu (Ancestor Unit) with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match
  nth u
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit)
with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end rewrite step3.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) =
nth u dsu (Ancestor Unit)
match nth u dsu (Ancestor Unit) with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match nth u dsu (Ancestor Unit) with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end
  remember (nth u dsu (Ancestor Unit)) as s eqn:hs.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: Slot
hs: s = nth u dsu (Ancestor Unit)
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = s
match s with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match s with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end symmetry in hs.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: Slot
hs: nth u dsu (Ancestor Unit) = s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = s
match s with
| ReferTo _0 => u :: ancestorChain dsu fuel _0
| Ancestor _ => [u]
end =
match s with
| ReferTo _0 =>
    u
    :: ancestorChain
         (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
            dsu) fuel _0
| Ancestor _ => [u]
end destruct s as [s | s]; [| reflexivity].
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
u :: ancestorChain dsu fuel s =
u
:: ancestorChain
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
     fuel s rewrite <- IH.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
u :: ancestorChain dsu fuel s =
u :: ancestorChain dsu fuel s
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
s < length dsu
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) s)
  → nth _0 (ancestorChain dsu (length dsu) s) 0 ≠ x {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
u :: ancestorChain dsu fuel s =
u :: ancestorChain dsu fuel s reflexivity. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
s < length dsu
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) s)
  → nth _0 (ancestorChain dsu (length dsu) s) 0 ≠ x
  -
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
s < length dsu exact (h1 u s hs).
  -
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) s)
  → nth _0 (ancestorChain dsu (length dsu) s) 0 ≠ x intros y hy.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x pose proof hd (S y) as step4.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y <
length (ancestorChain dsu (length dsu) u)
→ nth (S y) (ancestorChain dsu (length dsu) u)
    0 ≠ x
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x remember (ancestorChain dsu (length dsu) u) as toChop eqn:hC.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
toChop: list nat
hC: toChop = ancestorChain dsu (length dsu) u
hd: ∀ _0 : nat,
  _0 < length toChop → nth _0 toChop 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < length toChop → nth (S y) toChop 0 ≠ x
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x symmetry in hC.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
toChop: list nat
hC: ancestorChain dsu (length dsu) u = toChop
hd: ∀ _0 : nat,
  _0 < length toChop → nth _0 toChop 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < length toChop → nth (S y) toChop 0 ≠ x
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x destruct toChop as [| head tail].
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hC: ancestorChain dsu (length dsu) u = []
hd: ∀ _0 : nat, _0 < length [] → nth _0 [] 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < length [] → nth (S y) [] 0 ≠ x
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
head: nat
tail: list nat
hC: ancestorChain dsu (length dsu) u = head :: tail
hd: ∀ _0 : nat,
  _0 < length (head :: tail)
  → nth _0 (head :: tail) 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < length (head :: tail)
→ nth (S y) (head :: tail) 0 ≠ x
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
hC: ancestorChain dsu (length dsu) u = []
hd: ∀ _0 : nat, _0 < length [] → nth _0 [] 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < length [] → nth (S y) [] 0 ≠ x
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x destruct (length dsu); simpl in hC; [easy |].
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
_0: nat
h2: withoutCyclesN dsu (S _0)
h4: x < S _0
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _1 : nat,
  _1 < S _0
  → (∀ _2 : nat,
       _2 < length (ancestorChain dsu (S _0) _1)
       → nth _2 (ancestorChain dsu (S _0) _1) 0
         ≠ x)
    → ancestorChain dsu fuel _1 =
      ancestorChain
        (<[x:=ReferTo (ancestor dsu (S _0) x)]>
           dsu) fuel _1
u: nat
h3: u < S _0
hC: match nth u dsu (Ancestor Unit) with
| ReferTo _1 => u :: ancestorChain dsu _0 _1
| Ancestor _ => [u]
end = []
hd: ∀ _0 : nat, _0 < length [] → nth _0 [] 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (S _0) x)]> dsu
!! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (S _0) x)]> dsu)
  (Ancestor Unit) = ReferTo s
y: nat
hy: y < length (ancestorChain dsu (S _0) s)
step4: S y < length [] → nth (S y) [] 0 ≠ x
nth y (ancestorChain dsu (S _0) s) 0 ≠ x destruct (nth u dsu (Ancestor Unit)); simpl in hC; easy. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
head: nat
tail: list nat
hC: ancestorChain dsu (length dsu) u = head :: tail
hd: ∀ _0 : nat,
  _0 < length (head :: tail)
  → nth _0 (head :: tail) 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < length (head :: tail)
→ nth (S y) (head :: tail) 0 ≠ x
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    assert (hz : head = u).
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
head: nat
tail: list nat
hC: ancestorChain dsu (length dsu) u = head :: tail
hd: ∀ _0 : nat,
  _0 < length (head :: tail)
  → nth _0 (head :: tail) 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < length (head :: tail)
→ nth (S y) (head :: tail) 0 ≠ x
head = u
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
head: nat
tail: list nat
hC: ancestorChain dsu (length dsu) u = head :: tail
hd: ∀ _0 : nat,
  _0 < length (head :: tail)
  → nth _0 (head :: tail) 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < length (head :: tail)
→ nth (S y) (head :: tail) 0 ≠ x
hz: head = u
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
head: nat
tail: list nat
hC: ancestorChain dsu (length dsu) u = head :: tail
hd: ∀ _0 : nat,
  _0 < length (head :: tail)
  → nth _0 (head :: tail) 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < length (head :: tail)
→ nth (S y) (head :: tail) 0 ≠ x
head = u destruct (length dsu); simpl in hC; [easy |].
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
_0: nat
h2: withoutCyclesN dsu (S _0)
h4: x < S _0
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _1 : nat,
  _1 < S _0
  → (∀ _2 : nat,
       _2 < length (ancestorChain dsu (S _0) _1)
       → nth _2 (ancestorChain dsu (S _0) _1) 0
         ≠ x)
    → ancestorChain dsu fuel _1 =
      ancestorChain
        (<[x:=ReferTo (ancestor dsu (S _0) x)]>
           dsu) fuel _1
u: nat
h3: u < S _0
head: nat
tail: list nat
hC: match nth u dsu (Ancestor Unit) with
| ReferTo _1 => u :: ancestorChain dsu _0 _1
| Ancestor _ => [u]
end = head :: tail
hd: ∀ _0 : nat,
  _0 < length (head :: tail)
  → nth _0 (head :: tail) 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (S _0) x)]> dsu
!! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (S _0) x)]> dsu)
  (Ancestor Unit) = ReferTo s
y: nat
hy: y < length (ancestorChain dsu (S _0) s)
step4: S y < length (head :: tail)
→ nth (S y) (head :: tail) 0 ≠ x
head = u destruct (nth u dsu (Ancestor Unit)); simpl in hC; try easy.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
_0: nat
h2: withoutCyclesN dsu (S _0)
h4: x < S _0
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _1 : nat,
  _1 < S _0
  → (∀ _2 : nat,
       _2 < length (ancestorChain dsu (S _0) _1)
       → nth _2 (ancestorChain dsu (S _0) _1) 0
         ≠ x)
    → ancestorChain dsu fuel _1 =
      ancestorChain
        (<[x:=ReferTo (ancestor dsu (S _0) x)]>
           dsu) fuel _1
u: nat
h3: u < S _0
head: nat
tail: list nat
_1: nat
hC: u :: ancestorChain dsu _0 _1 = head :: tail
hd: ∀ _0 : nat,
  _0 < length (head :: tail)
  → nth _0 (head :: tail) 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (S _0) x)]> dsu
!! u = dsu !! u
s: nat
hs: ReferTo _1 = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (S _0) x)]> dsu)
  (Ancestor Unit) = ReferTo s
y: nat
hy: y < length (ancestorChain dsu (S _0) s)
step4: S y < length (head :: tail)
→ nth (S y) (head :: tail) 0 ≠ x
head = u inversion hC.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
_0: nat
h2: withoutCyclesN dsu (S _0)
h4: x < S _0
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _1 : nat,
  _1 < S _0
  → (∀ _2 : nat,
       _2 < length (ancestorChain dsu (S _0) _1)
       → nth _2 (ancestorChain dsu (S _0) _1) 0
         ≠ x)
    → ancestorChain dsu fuel _1 =
      ancestorChain
        (<[x:=ReferTo (ancestor dsu (S _0) x)]>
           dsu) fuel _1
u: nat
h3: u < S _0
head: nat
tail: list nat
_1: nat
hC: u :: ancestorChain dsu _0 _1 = head :: tail
hd: ∀ _0 : nat,
  _0 < length (head :: tail)
  → nth _0 (head :: tail) 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (S _0) x)]> dsu
!! u = dsu !! u
s: nat
hs: ReferTo _1 = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (S _0) x)]> dsu)
  (Ancestor Unit) = ReferTo s
y: nat
hy: y < length (ancestorChain dsu (S _0) s)
step4: S y < length (head :: tail)
→ nth (S y) (head :: tail) 0 ≠ x
_3: u = head
_4: ancestorChain dsu _0 _1 = tail
head = head reflexivity. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
head: nat
tail: list nat
hC: ancestorChain dsu (length dsu) u = head :: tail
hd: ∀ _0 : nat,
  _0 < length (head :: tail)
  → nth _0 (head :: tail) 0 ≠ x
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < length (head :: tail)
→ nth (S y) (head :: tail) 0 ≠ x
hz: head = u
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x subst head.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < length (u :: tail)
→ nth (S y) (u :: tail) 0 ≠ x
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x simpl in step4.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    pose proof validChainAncestorChain dsu (length dsu) u h3 h1 as [a [b c]].
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    assert (hL : 1 < length dsu).
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
1 < length dsu
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
1 < length dsu destruct (decide (length dsu = 0)) as [g | g].
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
g: length dsu = 0
1 < length dsu
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
g: length dsu ≠ 0
1 < length dsu
      {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
g: length dsu = 0
1 < length dsu pose proof nil_length_inv _ g.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
g: length dsu = 0
_0: dsu = []
1 < length dsu subst dsu.
fuel, x: nat
IH: ∀ _0 : nat,
  _0 < length []
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain [] (length []) _0)
       → nth _1 (ancestorChain [] (length []) _0)
           0 ≠ x)
    → ancestorChain [] fuel _0 =
      ancestorChain
        (<[x:=ReferTo (ancestor [] (length []) x)]>
           []) fuel _0
h5: match nth x [] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h4: x < length []
h2: withoutCyclesN [] (length [])
h1: noIllegalIndices []
u: nat
h3: u < length []
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain [] (length []) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor [] (length []) x)]> []
!! u = [] !! u
s: nat
step3: nth u
  (<[x:=ReferTo (ancestor [] (length []) x)]>
     []) (Ancestor Unit) = 
ReferTo s
hs: nth u [] (Ancestor Unit) = ReferTo s
y: nat
hy: y < length (ancestorChain [] (length []) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g: length [] = 0
c: ∀ _0 : nat,
  S _0 < length (ancestorChain [] (length []) u)
  → nth
      (nth _0 (ancestorChain [] (length []) u) 0)
      [] (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain [] (length []) u)
         0)
b: nth 0 (ancestorChain [] (length []) u) 0 <
length []
a: ancestorChain [] (length []) u ≠ []
1 < length [] simpl in *.
fuel, x: nat
IH: ∀ _0 : nat,
  _0 < 0
  → (∀ _1 : nat,
       _1 < 1
       → match _1 with
         | 0 => _0
         | _ => 0
         end ≠ x)
    → ancestorChain [] fuel _0 =
      ancestorChain (<[x:=ReferTo x]> []) fuel _0
h5: match
  match x with
  | 0 | _ => Ancestor Unit
  end
with
| ReferTo _ => true
| Ancestor _ => false
end
h4: x < 0
h2: withoutCyclesN [] 0
h1: noIllegalIndices []
u: nat
h3: u < 0
tail: list nat
hd: ∀ _0 : nat,
  _0 < S (length tail)
  → match _0 with
    | 0 => u
    | S _1 => nth _1 tail 0
    end ≠ x
hC: [u] = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo x]> [] !! u = [] !! u
s: nat
step3, hs: match u with
| 0 | _ => Ancestor Unit
end = ReferTo s
y: nat
hy: y < 1
step4: S y < S (length tail) → nth y tail 0 ≠ x
g: 0 = 0
c: ∀ _0 : nat,
  S _0 < 1
  → match match _0 with
          | 0 => u
          | _ => 0
          end with
    | 0 | _ => Ancestor Unit
    end = ReferTo match _0 with
                  | 0 | _ => 0
                  end
b: u < 0
a: [u] ≠ []
1 < 0 easy. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
g: length dsu ≠ 0
1 < length dsu
      destruct (decide (length dsu = 1)) as [g1 | g1].
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
g: length dsu ≠ 0
g1: length dsu = 1
1 < length dsu
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
g: length dsu ≠ 0
g1: length dsu ≠ 1
1 < length dsu
      {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
g: length dsu ≠ 0
g1: length dsu = 1
1 < length dsu destruct dsu as [| head taill].
fuel, x: nat
h1: noIllegalIndices []
h2: withoutCyclesN [] (length [])
h4: x < length []
h5: match nth x [] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length []
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain [] (length []) _0)
       → nth _1 (ancestorChain [] (length []) _0)
           0 ≠ x)
    → ancestorChain [] fuel _0 =
      ancestorChain
        (<[x:=ReferTo (ancestor [] (length []) x)]>
           []) fuel _0
u: nat
h3: u < length []
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain [] (length []) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor [] (length []) x)]> []
!! u = [] !! u
s: nat
hs: nth u [] (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor [] (length []) x)]>
     []) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain [] (length []) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain [] (length []) u ≠ []
b: nth 0 (ancestorChain [] (length []) u) 0 <
length []
c: ∀ _0 : nat,
  S _0 < length (ancestorChain [] (length []) u)
  → nth
      (nth _0 (ancestorChain [] (length []) u) 0)
      [] (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain [] (length []) u)
         0)
g: length [] ≠ 0
g1: length [] = 1
1 < length []
head: Slot
taill: list Slot
fuel, x: nat
h1: noIllegalIndices (head :: taill)
h2: withoutCyclesN (head :: taill)
  (length (head :: taill))
h4: x < length (head :: taill)
h5: match nth x (head :: taill) (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length (head :: taill)
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain (head :: taill)
            (length (head :: taill)) _0)
       → nth _1
           (ancestorChain 
              (head :: taill)
              (length (head :: taill)) _0) 0 ≠ x)
    → ancestorChain (head :: taill) fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor 
                   (head :: taill)
                   (length (head :: taill)) x)]>
           (head :: taill)) fuel _0
u: nat
h3: u < length (head :: taill)
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain (head :: taill)
  (length (head :: taill)) u = 
u :: tail
step1: u ≠ x
step2: <[x:=ReferTo
       (ancestor (head :: taill)
          (length (head :: taill)) x)]>
  (head :: taill) !! u = 
(head :: taill) !! u
s: nat
hs: nth u (head :: taill) (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo
          (ancestor (head :: taill)
             (length (head :: taill)) x)]>
     (head :: taill)) (Ancestor Unit) =
ReferTo s
y: nat
hy: y <
length
  (ancestorChain (head :: taill)
     (length (head :: taill)) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain (head :: taill)
  (length (head :: taill)) u ≠ []
b: nth 0
  (ancestorChain (head :: taill)
     (length (head :: taill)) u) 0 <
length (head :: taill)
c: ∀ _0 : nat,
  S _0 <
  length
    (ancestorChain (head :: taill)
       (length (head :: taill)) u)
  → nth
      (nth _0
         (ancestorChain (head :: taill)
            (length (head :: taill)) u) 0)
      (head :: taill) (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain (head :: taill)
            (length (head :: taill)) u) 0)
g: length (head :: taill) ≠ 0
g1: length (head :: taill) = 1
1 < length (head :: taill) {
fuel, x: nat
h1: noIllegalIndices []
h2: withoutCyclesN [] (length [])
h4: x < length []
h5: match nth x [] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length []
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain [] (length []) _0)
       → nth _1 (ancestorChain [] (length []) _0)
           0 ≠ x)
    → ancestorChain [] fuel _0 =
      ancestorChain
        (<[x:=ReferTo (ancestor [] (length []) x)]>
           []) fuel _0
u: nat
h3: u < length []
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain [] (length []) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor [] (length []) x)]> []
!! u = [] !! u
s: nat
hs: nth u [] (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor [] (length []) x)]>
     []) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain [] (length []) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain [] (length []) u ≠ []
b: nth 0 (ancestorChain [] (length []) u) 0 <
length []
c: ∀ _0 : nat,
  S _0 < length (ancestorChain [] (length []) u)
  → nth
      (nth _0 (ancestorChain [] (length []) u) 0)
      [] (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain [] (length []) u)
         0)
g: length [] ≠ 0
g1: length [] = 1
1 < length [] simpl in g1.
fuel, x: nat
h1: noIllegalIndices []
h2: withoutCyclesN [] (length [])
h4: x < length []
h5: match nth x [] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length []
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain [] (length []) _0)
       → nth _1 (ancestorChain [] (length []) _0)
           0 ≠ x)
    → ancestorChain [] fuel _0 =
      ancestorChain
        (<[x:=ReferTo (ancestor [] (length []) x)]>
           []) fuel _0
u: nat
h3: u < length []
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain [] (length []) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor [] (length []) x)]> []
!! u = [] !! u
s: nat
hs: nth u [] (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor [] (length []) x)]>
     []) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain [] (length []) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain [] (length []) u ≠ []
b: nth 0 (ancestorChain [] (length []) u) 0 <
length []
c: ∀ _0 : nat,
  S _0 < length (ancestorChain [] (length []) u)
  → nth
      (nth _0 (ancestorChain [] (length []) u) 0)
      [] (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain [] (length []) u)
         0)
g: length [] ≠ 0
g1: 0 = 1
1 < length [] easy. }
head: Slot
taill: list Slot
fuel, x: nat
h1: noIllegalIndices (head :: taill)
h2: withoutCyclesN (head :: taill)
  (length (head :: taill))
h4: x < length (head :: taill)
h5: match nth x (head :: taill) (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length (head :: taill)
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain (head :: taill)
            (length (head :: taill)) _0)
       → nth _1
           (ancestorChain 
              (head :: taill)
              (length (head :: taill)) _0) 0 ≠ x)
    → ancestorChain (head :: taill) fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor 
                   (head :: taill)
                   (length (head :: taill)) x)]>
           (head :: taill)) fuel _0
u: nat
h3: u < length (head :: taill)
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain (head :: taill)
  (length (head :: taill)) u = 
u :: tail
step1: u ≠ x
step2: <[x:=ReferTo
       (ancestor (head :: taill)
          (length (head :: taill)) x)]>
  (head :: taill) !! u = 
(head :: taill) !! u
s: nat
hs: nth u (head :: taill) (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo
          (ancestor (head :: taill)
             (length (head :: taill)) x)]>
     (head :: taill)) (Ancestor Unit) =
ReferTo s
y: nat
hy: y <
length
  (ancestorChain (head :: taill)
     (length (head :: taill)) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain (head :: taill)
  (length (head :: taill)) u ≠ []
b: nth 0
  (ancestorChain (head :: taill)
     (length (head :: taill)) u) 0 <
length (head :: taill)
c: ∀ _0 : nat,
  S _0 <
  length
    (ancestorChain (head :: taill)
       (length (head :: taill)) u)
  → nth
      (nth _0
         (ancestorChain (head :: taill)
            (length (head :: taill)) u) 0)
      (head :: taill) (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain (head :: taill)
            (length (head :: taill)) u) 0)
g: length (head :: taill) ≠ 0
g1: length (head :: taill) = 1
1 < length (head :: taill)
        simpl in g1.
head: Slot
taill: list Slot
fuel, x: nat
h1: noIllegalIndices (head :: taill)
h2: withoutCyclesN (head :: taill)
  (length (head :: taill))
h4: x < length (head :: taill)
h5: match nth x (head :: taill) (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length (head :: taill)
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain (head :: taill)
            (length (head :: taill)) _0)
       → nth _1
           (ancestorChain 
              (head :: taill)
              (length (head :: taill)) _0) 0 ≠ x)
    → ancestorChain (head :: taill) fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor 
                   (head :: taill)
                   (length (head :: taill)) x)]>
           (head :: taill)) fuel _0
u: nat
h3: u < length (head :: taill)
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain (head :: taill)
  (length (head :: taill)) u = 
u :: tail
step1: u ≠ x
step2: <[x:=ReferTo
       (ancestor (head :: taill)
          (length (head :: taill)) x)]>
  (head :: taill) !! u = 
(head :: taill) !! u
s: nat
hs: nth u (head :: taill) (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo
          (ancestor (head :: taill)
             (length (head :: taill)) x)]>
     (head :: taill)) (Ancestor Unit) =
ReferTo s
y: nat
hy: y <
length
  (ancestorChain (head :: taill)
     (length (head :: taill)) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain (head :: taill)
  (length (head :: taill)) u ≠ []
b: nth 0
  (ancestorChain (head :: taill)
     (length (head :: taill)) u) 0 <
length (head :: taill)
c: ∀ _0 : nat,
  S _0 <
  length
    (ancestorChain (head :: taill)
       (length (head :: taill)) u)
  → nth
      (nth _0
         (ancestorChain (head :: taill)
            (length (head :: taill)) u) 0)
      (head :: taill) (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain (head :: taill)
            (length (head :: taill)) u) 0)
g: length (head :: taill) ≠ 0
g1: S (length taill) = 1
1 < length (head :: taill) pose proof nil_length_inv taill ltac:(lia).
head: Slot
taill: list Slot
fuel, x: nat
h1: noIllegalIndices (head :: taill)
h2: withoutCyclesN (head :: taill)
  (length (head :: taill))
h4: x < length (head :: taill)
h5: match nth x (head :: taill) (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length (head :: taill)
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain (head :: taill)
            (length (head :: taill)) _0)
       → nth _1
           (ancestorChain 
              (head :: taill)
              (length (head :: taill)) _0) 0 ≠ x)
    → ancestorChain (head :: taill) fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor 
                   (head :: taill)
                   (length (head :: taill)) x)]>
           (head :: taill)) fuel _0
u: nat
h3: u < length (head :: taill)
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain (head :: taill)
  (length (head :: taill)) u = 
u :: tail
step1: u ≠ x
step2: <[x:=ReferTo
       (ancestor (head :: taill)
          (length (head :: taill)) x)]>
  (head :: taill) !! u = 
(head :: taill) !! u
s: nat
hs: nth u (head :: taill) (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo
          (ancestor (head :: taill)
             (length (head :: taill)) x)]>
     (head :: taill)) (Ancestor Unit) =
ReferTo s
y: nat
hy: y <
length
  (ancestorChain (head :: taill)
     (length (head :: taill)) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain (head :: taill)
  (length (head :: taill)) u ≠ []
b: nth 0
  (ancestorChain (head :: taill)
     (length (head :: taill)) u) 0 <
length (head :: taill)
c: ∀ _0 : nat,
  S _0 <
  length
    (ancestorChain (head :: taill)
       (length (head :: taill)) u)
  → nth
      (nth _0
         (ancestorChain (head :: taill)
            (length (head :: taill)) u) 0)
      (head :: taill) (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain (head :: taill)
            (length (head :: taill)) u) 0)
g: length (head :: taill) ≠ 0
g1: S (length taill) = 1
_0: taill = []
1 < length (head :: taill) subst taill.
head: Slot
fuel, x: nat
IH: ∀ _0 : nat,
  _0 < length [head]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [head] (length [head]) _0)
       → nth _1
           (ancestorChain [head] 
              (length [head]) _0) 0 ≠ x)
    → ancestorChain [head] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [head] 
                   (length [head]) x)]> [head])
        fuel _0
h5: match nth x [head] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h4: x < length [head]
h2: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
u: nat
h3: u < length [head]
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain [head] (length [head]) u =
u :: tail
step1: u ≠ x
step2: <[x:=ReferTo
       (ancestor [head] (length [head]) x)]>
  [head] !! u = [head] !! u
s: nat
step3: nth u
  (<[x:=ReferTo
          (ancestor [head] (length [head]) x)]>
     [head]) (Ancestor Unit) = 
ReferTo s
hs: nth u [head] (Ancestor Unit) = ReferTo s
y: nat
hy: y <
length (ancestorChain [head] (length [head]) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
g: length [head] ≠ 0
c: ∀ _0 : nat,
  S _0 <
  length (ancestorChain [head] (length [head]) u)
  → nth
      (nth _0
         (ancestorChain [head] (length [head]) u)
         0) [head] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [head] (length [head]) u)
         0)
b: nth 0 (ancestorChain [head] (length [head]) u) 0 <
length [head]
a: ancestorChain [head] (length [head]) u ≠ []
1 < length [head]
        pose proof (ltac:(simpl in *; lia) : u = 0).
head: Slot
fuel, x: nat
IH: ∀ _0 : nat,
  _0 < length [head]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [head] (length [head]) _0)
       → nth _1
           (ancestorChain [head] 
              (length [head]) _0) 0 ≠ x)
    → ancestorChain [head] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [head] 
                   (length [head]) x)]> [head])
        fuel _0
h5: match nth x [head] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h4: x < length [head]
h2: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
u: nat
h3: u < length [head]
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain [head] (length [head]) u =
u :: tail
step1: u ≠ x
step2: <[x:=ReferTo
       (ancestor [head] (length [head]) x)]>
  [head] !! u = [head] !! u
s: nat
step3: nth u
  (<[x:=ReferTo
          (ancestor [head] (length [head]) x)]>
     [head]) (Ancestor Unit) = 
ReferTo s
hs: nth u [head] (Ancestor Unit) = ReferTo s
y: nat
hy: y <
length (ancestorChain [head] (length [head]) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
g: length [head] ≠ 0
c: ∀ _0 : nat,
  S _0 <
  length (ancestorChain [head] (length [head]) u)
  → nth
      (nth _0
         (ancestorChain [head] (length [head]) u)
         0) [head] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [head] (length [head]) u)
         0)
b: nth 0 (ancestorChain [head] (length [head]) u) 0 <
length [head]
a: ancestorChain [head] (length [head]) u ≠ []
_0: u = 0
1 < length [head] subst u.
head: Slot
fuel, x: nat
IH: ∀ _0 : nat,
  _0 < length [head]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [head] (length [head]) _0)
       → nth _1
           (ancestorChain [head] 
              (length [head]) _0) 0 ≠ x)
    → ancestorChain [head] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [head] 
                   (length [head]) x)]> [head])
        fuel _0
h5: match nth x [head] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h4: x < length [head]
h2: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
h3: 0 < length [head]
tail: list nat
step2: <[x:=ReferTo
       (ancestor [head] (length [head]) x)]>
  [head] !! 0 = [head] !! 0
step1: 0 ≠ x
hC: ancestorChain [head] (length [head]) 0 =
0 :: tail
hd: ∀ _0 : nat,
  _0 < length (0 :: tail)
  → nth _0 (0 :: tail) 0 ≠ x
s: nat
hs: nth 0 [head] (Ancestor Unit) = ReferTo s
step3: nth 0
  (<[x:=ReferTo
          (ancestor [head] (length [head]) x)]>
     [head]) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y <
length (ancestorChain [head] (length [head]) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
g: length [head] ≠ 0
a: ancestorChain [head] (length [head]) 0 ≠ []
b: nth 0 (ancestorChain [head] (length [head]) 0) 0 <
length [head]
c: ∀ _0 : nat,
  S _0 <
  length (ancestorChain [head] (length [head]) 0)
  → nth
      (nth _0
         (ancestorChain [head] (length [head]) 0)
         0) [head] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [head] (length [head]) 0)
         0)
1 < length [head] pose proof h1 0 s hs as g2.
head: Slot
fuel, x: nat
IH: ∀ _0 : nat,
  _0 < length [head]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [head] (length [head]) _0)
       → nth _1
           (ancestorChain [head] 
              (length [head]) _0) 0 ≠ x)
    → ancestorChain [head] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [head] 
                   (length [head]) x)]> [head])
        fuel _0
h5: match nth x [head] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h4: x < length [head]
h2: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
h3: 0 < length [head]
tail: list nat
step2: <[x:=ReferTo
       (ancestor [head] (length [head]) x)]>
  [head] !! 0 = [head] !! 0
step1: 0 ≠ x
hC: ancestorChain [head] (length [head]) 0 =
0 :: tail
hd: ∀ _0 : nat,
  _0 < length (0 :: tail)
  → nth _0 (0 :: tail) 0 ≠ x
s: nat
hs: nth 0 [head] (Ancestor Unit) = ReferTo s
step3: nth 0
  (<[x:=ReferTo
          (ancestor [head] (length [head]) x)]>
     [head]) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y <
length (ancestorChain [head] (length [head]) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
g: length [head] ≠ 0
a: ancestorChain [head] (length [head]) 0 ≠ []
b: nth 0 (ancestorChain [head] (length [head]) 0) 0 <
length [head]
c: ∀ _0 : nat,
  S _0 <
  length (ancestorChain [head] (length [head]) 0)
  → nth
      (nth _0
         (ancestorChain [head] (length [head]) 0)
         0) [head] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [head] (length [head]) 0)
         0)
g2: s < length [head]
1 < length [head] simpl in g2.
head: Slot
fuel, x: nat
IH: ∀ _0 : nat,
  _0 < length [head]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [head] (length [head]) _0)
       → nth _1
           (ancestorChain [head] 
              (length [head]) _0) 0 ≠ x)
    → ancestorChain [head] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [head] 
                   (length [head]) x)]> [head])
        fuel _0
h5: match nth x [head] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h4: x < length [head]
h2: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
h3: 0 < length [head]
tail: list nat
step2: <[x:=ReferTo
       (ancestor [head] (length [head]) x)]>
  [head] !! 0 = [head] !! 0
step1: 0 ≠ x
hC: ancestorChain [head] (length [head]) 0 =
0 :: tail
hd: ∀ _0 : nat,
  _0 < length (0 :: tail)
  → nth _0 (0 :: tail) 0 ≠ x
s: nat
hs: nth 0 [head] (Ancestor Unit) = ReferTo s
step3: nth 0
  (<[x:=ReferTo
          (ancestor [head] (length [head]) x)]>
     [head]) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y <
length (ancestorChain [head] (length [head]) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
g: length [head] ≠ 0
a: ancestorChain [head] (length [head]) 0 ≠ []
b: nth 0 (ancestorChain [head] (length [head]) 0) 0 <
length [head]
c: ∀ _0 : nat,
  S _0 <
  length (ancestorChain [head] (length [head]) 0)
  → nth
      (nth _0
         (ancestorChain [head] (length [head]) 0)
         0) [head] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [head] (length [head]) 0)
         0)
g2: s < 1
1 < length [head]
        pose proof (ltac:(lia) : s = 0).
head: Slot
fuel, x: nat
IH: ∀ _0 : nat,
  _0 < length [head]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [head] (length [head]) _0)
       → nth _1
           (ancestorChain [head] 
              (length [head]) _0) 0 ≠ x)
    → ancestorChain [head] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [head] 
                   (length [head]) x)]> [head])
        fuel _0
h5: match nth x [head] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h4: x < length [head]
h2: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
h3: 0 < length [head]
tail: list nat
step2: <[x:=ReferTo
       (ancestor [head] (length [head]) x)]>
  [head] !! 0 = [head] !! 0
step1: 0 ≠ x
hC: ancestorChain [head] (length [head]) 0 =
0 :: tail
hd: ∀ _0 : nat,
  _0 < length (0 :: tail)
  → nth _0 (0 :: tail) 0 ≠ x
s: nat
hs: nth 0 [head] (Ancestor Unit) = ReferTo s
step3: nth 0
  (<[x:=ReferTo
          (ancestor [head] (length [head]) x)]>
     [head]) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y <
length (ancestorChain [head] (length [head]) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
g: length [head] ≠ 0
a: ancestorChain [head] (length [head]) 0 ≠ []
b: nth 0 (ancestorChain [head] (length [head]) 0) 0 <
length [head]
c: ∀ _0 : nat,
  S _0 <
  length (ancestorChain [head] (length [head]) 0)
  → nth
      (nth _0
         (ancestorChain [head] (length [head]) 0)
         0) [head] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [head] (length [head]) 0)
         0)
g2: s < 1
_0: s = 0
1 < length [head] subst s.
head: Slot
fuel, x: nat
IH: ∀ _0 : nat,
  _0 < length [head]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [head] (length [head]) _0)
       → nth _1
           (ancestorChain [head] 
              (length [head]) _0) 0 ≠ x)
    → ancestorChain [head] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [head] 
                   (length [head]) x)]> [head])
        fuel _0
h5: match nth x [head] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h4: x < length [head]
h2: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
h3: 0 < length [head]
tail: list nat
step2: <[x:=ReferTo
       (ancestor [head] (length [head]) x)]>
  [head] !! 0 = [head] !! 0
step1: 0 ≠ x
hC: ancestorChain [head] (length [head]) 0 =
0 :: tail
hd: ∀ _0 : nat,
  _0 < length (0 :: tail)
  → nth _0 (0 :: tail) 0 ≠ x
step3: nth 0
  (<[x:=ReferTo
          (ancestor [head] (length [head]) x)]>
     [head]) (Ancestor Unit) = 
ReferTo 0
hs: nth 0 [head] (Ancestor Unit) = ReferTo 0
y: nat
hy: y <
length (ancestorChain [head] (length [head]) 0)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
g: length [head] ≠ 0
a: ancestorChain [head] (length [head]) 0 ≠ []
b: nth 0 (ancestorChain [head] (length [head]) 0) 0 <
length [head]
c: ∀ _0 : nat,
  S _0 <
  length (ancestorChain [head] (length [head]) 0)
  → nth
      (nth _0
         (ancestorChain [head] (length [head]) 0)
         0) [head] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [head] (length [head]) 0)
         0)
g2: 0 < 1
1 < length [head] pose proof h2 0 ltac:(simpl; lia) as g3.
head: Slot
fuel, x: nat
IH: ∀ _0 : nat,
  _0 < length [head]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [head] (length [head]) _0)
       → nth _1
           (ancestorChain [head] 
              (length [head]) _0) 0 ≠ x)
    → ancestorChain [head] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [head] 
                   (length [head]) x)]> [head])
        fuel _0
h5: match nth x [head] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h4: x < length [head]
h2: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
h3: 0 < length [head]
tail: list nat
step2: <[x:=ReferTo
       (ancestor [head] (length [head]) x)]>
  [head] !! 0 = [head] !! 0
step1: 0 ≠ x
hC: ancestorChain [head] (length [head]) 0 =
0 :: tail
hd: ∀ _0 : nat,
  _0 < length (0 :: tail)
  → nth _0 (0 :: tail) 0 ≠ x
step3: nth 0
  (<[x:=ReferTo
          (ancestor [head] (length [head]) x)]>
     [head]) (Ancestor Unit) = 
ReferTo 0
hs: nth 0 [head] (Ancestor Unit) = ReferTo 0
y: nat
hy: y <
length (ancestorChain [head] (length [head]) 0)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
g: length [head] ≠ 0
a: ancestorChain [head] (length [head]) 0 ≠ []
b: nth 0 (ancestorChain [head] (length [head]) 0) 0 <
length [head]
c: ∀ _0 : nat,
  S _0 <
  length (ancestorChain [head] (length [head]) 0)
  → nth
      (nth _0
         (ancestorChain [head] (length [head]) 0)
         0) [head] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [head] (length [head]) 0)
         0)
g2: 0 < 1
g3: match
  nth (ancestor [head] (length [head]) 0) [head]
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
1 < length [head] simpl in hs.
head: Slot
fuel, x: nat
IH: ∀ _0 : nat,
  _0 < length [head]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [head] (length [head]) _0)
       → nth _1
           (ancestorChain [head] 
              (length [head]) _0) 0 ≠ x)
    → ancestorChain [head] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [head] 
                   (length [head]) x)]> [head])
        fuel _0
h5: match nth x [head] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h4: x < length [head]
h2: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
h3: 0 < length [head]
tail: list nat
step2: <[x:=ReferTo
       (ancestor [head] (length [head]) x)]>
  [head] !! 0 = [head] !! 0
step1: 0 ≠ x
hC: ancestorChain [head] (length [head]) 0 =
0 :: tail
hd: ∀ _0 : nat,
  _0 < length (0 :: tail)
  → nth _0 (0 :: tail) 0 ≠ x
step3: nth 0
  (<[x:=ReferTo
          (ancestor [head] (length [head]) x)]>
     [head]) (Ancestor Unit) = 
ReferTo 0
hs: head = ReferTo 0
y: nat
hy: y <
length (ancestorChain [head] (length [head]) 0)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
g: length [head] ≠ 0
a: ancestorChain [head] (length [head]) 0 ≠ []
b: nth 0 (ancestorChain [head] (length [head]) 0) 0 <
length [head]
c: ∀ _0 : nat,
  S _0 <
  length (ancestorChain [head] (length [head]) 0)
  → nth
      (nth _0
         (ancestorChain [head] (length [head]) 0)
         0) [head] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [head] (length [head]) 0)
         0)
g2: 0 < 1
g3: match
  nth (ancestor [head] (length [head]) 0) [head]
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
1 < length [head] subst head.
fuel, x: nat
h3: 0 < length [ReferTo 0]
h1: noIllegalIndices [ReferTo 0]
h2: withoutCyclesN [ReferTo 0] (length [ReferTo 0])
h4: x < length [ReferTo 0]
h5: match nth x [ReferTo 0] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length [ReferTo 0]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [
            ReferTo 0] (length [ReferTo 0]) _0)
       → nth _1
           (ancestorChain [
              ReferTo 0] 
              (length [ReferTo 0]) _0) 0 ≠ x)
    → ancestorChain [ReferTo 0] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [
                   ReferTo 0]
                   (length [ReferTo 0]) x)]>
           [ReferTo 0]) fuel _0
tail: list nat
step2: <[x:=ReferTo
       (ancestor [ReferTo 0]
          (length [ReferTo 0]) x)]>
  [ReferTo 0] !! 0 = [ReferTo 0] !! 0
step1: 0 ≠ x
hC: ancestorChain [ReferTo 0] (length [ReferTo 0]) 0 =
0 :: tail
hd: ∀ _0 : nat,
  _0 < length (0 :: tail)
  → nth _0 (0 :: tail) 0 ≠ x
step3: nth 0
  (<[x:=ReferTo
          (ancestor [ReferTo 0]
             (length [ReferTo 0]) x)]>
     [ReferTo 0]) (Ancestor Unit) = 
ReferTo 0
y: nat
hy: y <
length
  (ancestorChain [ReferTo 0] 
     (length [ReferTo 0]) 0)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
c: ∀ _0 : nat,
  S _0 <
  length
    (ancestorChain [ReferTo 0]
       (length [ReferTo 0]) 0)
  → nth
      (nth _0
         (ancestorChain [
            ReferTo 0] (length [ReferTo 0]) 0) 0)
      [ReferTo 0] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [
            ReferTo 0] (length [ReferTo 0]) 0) 0)
b: nth 0
  (ancestorChain [ReferTo 0] 
     (length [ReferTo 0]) 0) 0 <
length [ReferTo 0]
a: ancestorChain [ReferTo 0] (length [ReferTo 0]) 0
≠ []
g: length [ReferTo 0] ≠ 0
g2: 0 < 1
g3: match
  nth
    (ancestor [ReferTo 0] (length [ReferTo 0]) 0)
    [ReferTo 0] (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
1 < length [ReferTo 0] simpl in g3.
fuel, x: nat
h3: 0 < length [ReferTo 0]
h1: noIllegalIndices [ReferTo 0]
h2: withoutCyclesN [ReferTo 0] (length [ReferTo 0])
h4: x < length [ReferTo 0]
h5: match nth x [ReferTo 0] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length [ReferTo 0]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [
            ReferTo 0] (length [ReferTo 0]) _0)
       → nth _1
           (ancestorChain [
              ReferTo 0] 
              (length [ReferTo 0]) _0) 0 ≠ x)
    → ancestorChain [ReferTo 0] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [
                   ReferTo 0]
                   (length [ReferTo 0]) x)]>
           [ReferTo 0]) fuel _0
tail: list nat
step2: <[x:=ReferTo
       (ancestor [ReferTo 0]
          (length [ReferTo 0]) x)]>
  [ReferTo 0] !! 0 = [ReferTo 0] !! 0
step1: 0 ≠ x
hC: ancestorChain [ReferTo 0] (length [ReferTo 0]) 0 =
0 :: tail
hd: ∀ _0 : nat,
  _0 < length (0 :: tail)
  → nth _0 (0 :: tail) 0 ≠ x
step3: nth 0
  (<[x:=ReferTo
          (ancestor [ReferTo 0]
             (length [ReferTo 0]) x)]>
     [ReferTo 0]) (Ancestor Unit) = 
ReferTo 0
y: nat
hy: y <
length
  (ancestorChain [ReferTo 0] 
     (length [ReferTo 0]) 0)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
c: ∀ _0 : nat,
  S _0 <
  length
    (ancestorChain [ReferTo 0]
       (length [ReferTo 0]) 0)
  → nth
      (nth _0
         (ancestorChain [
            ReferTo 0] (length [ReferTo 0]) 0) 0)
      [ReferTo 0] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [
            ReferTo 0] (length [ReferTo 0]) 0) 0)
b: nth 0
  (ancestorChain [ReferTo 0] 
     (length [ReferTo 0]) 0) 0 <
length [ReferTo 0]
a: ancestorChain [ReferTo 0] (length [ReferTo 0]) 0
≠ []
g: length [ReferTo 0] ≠ 0
g2: 0 < 1
g3: False
1 < length [ReferTo 0] exfalso.
fuel, x: nat
h3: 0 < length [ReferTo 0]
h1: noIllegalIndices [ReferTo 0]
h2: withoutCyclesN [ReferTo 0] (length [ReferTo 0])
h4: x < length [ReferTo 0]
h5: match nth x [ReferTo 0] (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length [ReferTo 0]
  → (∀ _1 : nat,
       _1 <
       length
         (ancestorChain [
            ReferTo 0] (length [ReferTo 0]) _0)
       → nth _1
           (ancestorChain [
              ReferTo 0] 
              (length [ReferTo 0]) _0) 0 ≠ x)
    → ancestorChain [ReferTo 0] fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor [
                   ReferTo 0]
                   (length [ReferTo 0]) x)]>
           [ReferTo 0]) fuel _0
tail: list nat
step2: <[x:=ReferTo
       (ancestor [ReferTo 0]
          (length [ReferTo 0]) x)]>
  [ReferTo 0] !! 0 = [ReferTo 0] !! 0
step1: 0 ≠ x
hC: ancestorChain [ReferTo 0] (length [ReferTo 0]) 0 =
0 :: tail
hd: ∀ _0 : nat,
  _0 < length (0 :: tail)
  → nth _0 (0 :: tail) 0 ≠ x
step3: nth 0
  (<[x:=ReferTo
          (ancestor [ReferTo 0]
             (length [ReferTo 0]) x)]>
     [ReferTo 0]) (Ancestor Unit) = 
ReferTo 0
y: nat
hy: y <
length
  (ancestorChain [ReferTo 0] 
     (length [ReferTo 0]) 0)
step4: S y < S (length tail) → nth y tail 0 ≠ x
g1: S (length []) = 1
c: ∀ _0 : nat,
  S _0 <
  length
    (ancestorChain [ReferTo 0]
       (length [ReferTo 0]) 0)
  → nth
      (nth _0
         (ancestorChain [
            ReferTo 0] (length [ReferTo 0]) 0) 0)
      [ReferTo 0] (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain [
            ReferTo 0] (length [ReferTo 0]) 0) 0)
b: nth 0
  (ancestorChain [ReferTo 0] 
     (length [ReferTo 0]) 0) 0 <
length [ReferTo 0]
a: ancestorChain [ReferTo 0] (length [ReferTo 0]) 0
≠ []
g: length [ReferTo 0] ≠ 0
g2: 0 < 1
g3: False
False exact g3. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
g: length dsu ≠ 0
g1: length dsu ≠ 1
1 < length dsu lia. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    assert (hA : 1 < length (ancestorChain dsu (length dsu) u)).
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
1 < length (ancestorChain dsu (length dsu) u)
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
1 < length (ancestorChain dsu (length dsu) u) assert (hS : exists x, length dsu = S (S (x))).
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
∃ _0 : nat, length dsu = S (S _0)
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hS: ∃ _0 : nat, length dsu = S (S _0)
1 < length (ancestorChain dsu (length dsu) u) {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
∃ _0 : nat, length dsu = S (S _0) exists (length dsu - 2).
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
length dsu = S (S (length dsu - 2)) lia. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hS: ∃ _0 : nat, length dsu = S (S _0)
1 < length (ancestorChain dsu (length dsu) u)
      destruct hS as [v vv].
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
v: nat
vv: length dsu = S (S v)
1 < length (ancestorChain dsu (length dsu) u) rewrite vv.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
v: nat
vv: length dsu = S (S v)
1 < length (ancestorChain dsu (S (S v)) u) simpl.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
v: nat
vv: length dsu = S (S v)
1 <
length
  match nth u dsu (Ancestor Unit) with
  | ReferTo _0 =>
      u
      :: match nth _0 dsu (Ancestor Unit) with
         | ReferTo _1 => _0 :: ancestorChain dsu v _1
         | Ancestor _ => [_0]
         end
  | Ancestor _ => [u]
  end rewrite hs.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
v: nat
vv: length dsu = S (S v)
1 <
length
  (u
   :: match nth s dsu (Ancestor Unit) with
      | ReferTo _0 => s :: ancestorChain dsu v _0
      | Ancestor _ => [s]
      end) destruct (nth s dsu (Ancestor Unit)); simpl; lia. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    pose proof c 0 hA as hQ.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: nth (nth 0 (ancestorChain dsu (length dsu) u) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth 1 (ancestorChain dsu (length dsu) u) 0)
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x rewrite hC in hQ.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: nth (nth 0 (u :: tail) 0) dsu (Ancestor Unit) =
ReferTo (nth 1 (u :: tail) 0)
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x simpl in hQ.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: nth u dsu (Ancestor Unit) =
ReferTo (nth 0 tail 0)
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x rewrite hs in hQ.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: ReferTo s = ReferTo (nth 0 tail 0)
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x injection hQ.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: ReferTo s = ReferTo (nth 0 tail 0)
s = nth 0 tail 0
→ nth y (ancestorChain dsu (length dsu) s) 0 ≠ x clear hQ.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
s = nth 0 tail 0
→ nth y (ancestorChain dsu (length dsu) s) 0 ≠ x intro hQ.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    pose proof (fun x => validChainAncestorLength dsu tail h1 x s ltac:(symmetry; exact hQ)) as step5.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    assert (hh : 0 < length tail).
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
0 < length tail
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
0 < length tail rewrite hC in hA.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (u :: tail)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
0 < length tail simpl in hA.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < S (length tail)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
0 < length tail lia. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    assert (step6 : validChainToAncestor dsu tail).
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
validChainToAncestor dsu tail
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
step6: validChainToAncestor dsu tail
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
    {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
validChainToAncestor dsu tail repeat split.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
tail ≠ []
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
nth 0 tail 0 < length dsu
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
∀ _0 : nat,
  S _0 < length tail
  → nth (nth _0 tail 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) tail 0)
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0
      -
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
tail ≠ [] rewrite hC in hA.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (u :: tail)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
tail ≠ [] simpl in hA.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < S (length tail)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
tail ≠ [] intro cc.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < S (length tail)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
cc: tail = []
False rewrite cc in hA.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < S (length [])
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
cc: tail = []
False simpl in hA.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < 1
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
cc: tail = []
False lia.
      -
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
nth 0 tail 0 < length dsu rewrite <- hQ.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
s < length dsu exact (h1 u s hs).
      -
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
∀ _0 : nat,
  S _0 < length tail
  → nth (nth _0 tail 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) tail 0) intros g1 g2.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g1: nat
g2: S g1 < length tail
nth (nth g1 tail 0) dsu (Ancestor Unit) =
ReferTo (nth (S g1) tail 0) rewrite hC in c.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (u :: tail)
  → nth (nth _0 (u :: tail) 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) (u :: tail) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g1: nat
g2: S g1 < length tail
nth (nth g1 tail 0) dsu (Ancestor Unit) =
ReferTo (nth (S g1) tail 0) pose proof c (S g1) as d.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (u :: tail)
  → nth (nth _0 (u :: tail) 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) (u :: tail) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g1: nat
g2: S g1 < length tail
d: S (S g1) < length (u :: tail)
→ nth (nth (S g1) (u :: tail) 0) dsu
    (Ancestor Unit) =
  ReferTo (nth (S (S g1)) (u :: tail) 0)
nth (nth g1 tail 0) dsu (Ancestor Unit) =
ReferTo (nth (S g1) tail 0) simpl in d.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (u :: tail)
  → nth (nth _0 (u :: tail) 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) (u :: tail) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g1: nat
g2: S g1 < length tail
d: S (S g1) < S (length tail)
→ nth (nth g1 tail 0) dsu (Ancestor Unit) =
  ReferTo (nth (S g1) tail 0)
nth (nth g1 tail 0) dsu (Ancestor Unit) =
ReferTo (nth (S g1) tail 0) apply d.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (u :: tail)
  → nth (nth _0 (u :: tail) 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) (u :: tail) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g1: nat
g2: S g1 < length tail
d: S (S g1) < S (length tail)
→ nth (nth g1 tail 0) dsu (Ancestor Unit) =
  ReferTo (nth (S g1) tail 0)
S (S g1) < S (length tail) lia.
      -
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0 pose proof h2 u h3 as g3.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0 pose proof (ltac:(exists (nth (ancestor dsu (length dsu) u) dsu (Ancestor Unit)); reflexivity) : exists e, nth (ancestor dsu (length dsu) u) dsu (Ancestor Unit) = e) as [[e | e] he].
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
e: nat
he: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = ReferTo e
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
e: Tree
he: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = Ancestor e
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0 {
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
e: nat
he: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = ReferTo e
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0 rewrite he in g3.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: false
e: nat
he: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = ReferTo e
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0 exfalso.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: false
e: nat
he: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = ReferTo e
False exact g3. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
e: Tree
he: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = Ancestor e
∃ _0 : Tree,
  nth (nth (length tail - 1) tail 0) dsu
    (Ancestor Unit) = Ancestor _0 exists e.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
e: Tree
he: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = Ancestor e
nth (nth (length tail - 1) tail 0) dsu (Ancestor Unit) =
Ancestor e rewrite <- ancestorEqLastAncestorChain in he.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
e: Tree
he: nth
  (nth
     (length (ancestorChain dsu (length dsu) u) -
      1) (ancestorChain dsu (length dsu) u) 0)
  dsu (Ancestor Unit) = Ancestor e
nth (nth (length tail - 1) tail 0) dsu (Ancestor Unit) =
Ancestor e rewrite hC in he.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
e: Tree
he: nth (nth (length (u :: tail) - 1) (u :: tail) 0)
  dsu (Ancestor Unit) = Ancestor e
nth (nth (length tail - 1) tail 0) dsu (Ancestor Unit) =
Ancestor e rewrite (ltac:(simpl; lia) : length (u :: tail) - 1 = length tail) in he.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
e: Tree
he: nth (nth (length tail) (u :: tail) 0) dsu
  (Ancestor Unit) = Ancestor e
nth (nth (length tail - 1) tail 0) dsu (Ancestor Unit) =
Ancestor e rewrite (ltac:(lia) : length tail = S (length tail - 1)) in he.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
e: Tree
he: nth (nth (S (length tail - 1)) (u :: tail) 0) dsu
  (Ancestor Unit) = Ancestor e
nth (nth (length tail - 1) tail 0) dsu (Ancestor Unit) =
Ancestor e simpl in he.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
g3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
e: Tree
he: nth (nth (length tail - 1) tail 0) dsu
  (Ancestor Unit) = Ancestor e
nth (nth (length tail - 1) tail 0) dsu (Ancestor Unit) =
Ancestor e exact he. }
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
step6: validChainToAncestor dsu tail
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x
      pose proof step5 step6 as step7.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length (ancestorChain dsu (length dsu) s)
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail
→ tail = ancestorChain dsu (length dsu) s
hh: 0 < length tail
step6: validChainToAncestor dsu tail
step7: tail = ancestorChain dsu (length dsu) s
nth y (ancestorChain dsu (length dsu) s) 0 ≠ x rewrite <- step7 in *.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length tail
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail → tail = tail
hh: 0 < length tail
step6: validChainToAncestor dsu tail
step7: tail = ancestorChain dsu (length dsu) s
nth y tail 0 ≠ x apply step4.
dsu: list Slot
fuel, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
IH: ∀ _0 : nat,
  _0 < length dsu
  → (∀ _1 : nat,
       _1 <
       length (ancestorChain dsu (length dsu) _0)
       → nth _1
           (ancestorChain dsu (length dsu) _0) 0
         ≠ x)
    → ancestorChain dsu fuel _0 =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu) fuel _0
u: nat
h3: u < length dsu
tail: list nat
hd: ∀ _0 : nat,
  _0 < length (u :: tail)
  → nth _0 (u :: tail) 0 ≠ x
hC: ancestorChain dsu (length dsu) u = u :: tail
step1: u ≠ x
step2: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! u = dsu !! u
s: nat
hs: nth u dsu (Ancestor Unit) = ReferTo s
step3: nth u
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo s
y: nat
hy: y < length tail
step4: S y < S (length tail) → nth y tail 0 ≠ x
a: ancestorChain dsu (length dsu) u ≠ []
b: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
c: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
hL: 1 < length dsu
hA: 1 < length (ancestorChain dsu (length dsu) u)
hQ: s = nth 0 tail 0
step5: validChainToAncestor dsu tail → tail = tail
hh: 0 < length tail
step6: validChainToAncestor dsu tail
step7: tail = ancestorChain dsu (length dsu) s
S y < S (length tail) lia.
Qed.

Lemma ancestorChainInsertPresent (dsu : list Slot) chain (u x : nat) (h0 : validChainToAncestor dsu chain) (hV : nth 0 chain 0 = u) (h1 : noIllegalIndices dsu) (h2 : withoutCyclesN dsu (length dsu)) (h3 : u < length dsu) (h4 : x < length dsu) (h5 : match nth x dsu (Ancestor Unit) with | ReferTo _ => true | Ancestor _ => false end) i (hd : i < length chain) (he : nth i chain 0 = x) (t : nat) (hT : nth t chain 0 = ancestor dsu (length dsu) x) (hT1 : i < t) (hT2 : t < length chain) : take (S i) chain ++ drop t chain = ancestorChain (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) (length ((<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu))) u.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
take (S i) chain ++ drop t chain =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu))
  u
Proof.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
take (S i) chain ++ drop t chain =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu))
  u
  assert (h1' : noIllegalIndices (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
take (S i) chain ++ drop t chain =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu))
  u
  {
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) intros a b c.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: nth a
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo b
b <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    destruct (decide (a = x)) as [h | h].
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: nth a
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo b
h: a = x
b <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: nth a
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo b
h: a ≠ x
b <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    -
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: nth a
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo b
h: a = x
b <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) rewrite h in c.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: nth x
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo b
h: a = x
b <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) rewrite nth_lookup, list_lookup_insert in c; [| lia].
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: default (Ancestor Unit)
  (Some (ReferTo (ancestor dsu (length dsu) x))) =
ReferTo b
h: a = x
b <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) simpl in c.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: ReferTo (ancestor dsu (length dsu) x) = ReferTo b
h: a = x
b <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) rewrite insert_length.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: ReferTo (ancestor dsu (length dsu) x) = ReferTo b
h: a = x
b < length dsu injection c.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: ReferTo (ancestor dsu (length dsu) x) = ReferTo b
h: a = x
ancestor dsu (length dsu) x = b → b < length dsu intro d.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: ReferTo (ancestor dsu (length dsu) x) = ReferTo b
h: a = x
d: ancestor dsu (length dsu) x = b
b < length dsu subst b.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a: nat
c: ReferTo (ancestor dsu (length dsu) x) =
ReferTo (ancestor dsu (length dsu) x)
h: a = x
ancestor dsu (length dsu) x < length dsu apply ancestorLtLength; assumption.
    -
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: nth a
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) = 
ReferTo b
h: a ≠ x
b <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) rewrite nth_lookup, list_lookup_insert_ne in c; [| lia].
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: default (Ancestor Unit) (dsu !! a) = ReferTo b
h: a ≠ x
b <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) rewrite <- nth_lookup in c.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: nth a dsu (Ancestor Unit) = ReferTo b
h: a ≠ x
b <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) rewrite insert_length.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
a, b: nat
c: nth a dsu (Ancestor Unit) = ReferTo b
h: a ≠ x
b < length dsu exact (h1 a b c). }
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
take (S i) chain ++ drop t chain =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu))
  u
  pose proof validChainAncestorLength (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) (take (S i) chain ++ drop t chain) h1' as step.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
take (S i) chain ++ drop t chain =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu))
  u
  assert (step1 : nth 0 (take (S i) chain ++ drop t chain) 0 = u).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
nth 0 (take (S i) chain ++ drop t chain) 0 = u
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
take (S i) chain ++ drop t chain =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu))
  u
  {
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
nth 0 (take (S i) chain ++ drop t chain) 0 = u rewrite app_nth1; [| rewrite take_length; lia].
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
nth 0 (take (S i) chain) 0 = u destruct chain as [| head tail].
dsu: list Slot
u, x: nat
h0: validChainToAncestor dsu []
hV: nth 0 [] 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length []
he: nth i [] 0 = x
t: nat
hT: nth t [] 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length []
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) [] ++ drop t [])
→ ∀ _0 : nat,
    nth 0 (take (S i) [] ++ drop t []) 0 = _0
    → take (S i) [] ++ drop t [] =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
nth 0 (take (S i) []) 0 = u
dsu: list Slot
head: nat
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (head :: tail)
hV: nth 0 (head :: tail) 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length (head :: tail)
he: nth i (head :: tail) 0 = x
t: nat
hT: nth t (head :: tail) 0 =
ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (head :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (head :: tail) ++
   drop t (head :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (head :: tail) ++
       drop t (head :: tail)) 0 = _0
    → take (S i) (head :: tail) ++
      drop t (head :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
nth 0 (take (S i) (head :: tail)) 0 = u
    -
dsu: list Slot
u, x: nat
h0: validChainToAncestor dsu []
hV: nth 0 [] 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length []
he: nth i [] 0 = x
t: nat
hT: nth t [] 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length []
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) [] ++ drop t [])
→ ∀ _0 : nat,
    nth 0 (take (S i) [] ++ drop t []) 0 = _0
    → take (S i) [] ++ drop t [] =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
nth 0 (take (S i) []) 0 = u cbv in h0.
dsu: list Slot
u, x: nat
h0: (([] = [] → False)
 ∧ 1
   ≤ (fix _0 (_1 : list Slot) : nat :=
        match _1 with
        | [] => 0
        | _ :: _3 => S (_0 _3)
        end) dsu
   ∧ ∀ _0 : nat,
       S (S _0) ≤ 0
       → (fix _1
            (_2 : nat) (_3 : list Slot)
            (_4 : Slot) {struct _3} : Slot :=
            match _2 with
            | 0 =>
                match _3 with
                | [] => _4
                | _5 :: _ => _5
                end
            | S _5 =>
                match _3 with
                | [] => _4
                | _ :: _7 => _1 _5 _7 _4
                end
            end) match _0 with
                 | 0 | _ => 0
                 end dsu 
           (Ancestor Unit) = 
         ReferTo 0)
∧ ∃ _0 : Tree,
    (fix _1
       (_2 : nat) (_3 : list Slot) 
       (_4 : Slot) {struct _3} : Slot :=
       match _2 with
       | 0 =>
           match _3 with
           | [] => _4
           | _5 :: _ => _5
           end
       | S _5 =>
           match _3 with
           | [] => _4
           | _ :: _7 => _1 _5 _7 _4
           end
       end) 0 dsu (Ancestor Unit) = 
    Ancestor _0
hV: nth 0 [] 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length []
he: nth i [] 0 = x
t: nat
hT: nth t [] 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length []
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) [] ++ drop t [])
→ ∀ _0 : nat,
    nth 0 (take (S i) [] ++ drop t []) 0 = _0
    → take (S i) [] ++ drop t [] =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
nth 0 (take (S i) []) 0 = u easy.
    -
dsu: list Slot
head: nat
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (head :: tail)
hV: nth 0 (head :: tail) 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length (head :: tail)
he: nth i (head :: tail) 0 = x
t: nat
hT: nth t (head :: tail) 0 =
ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (head :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (head :: tail) ++
   drop t (head :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (head :: tail) ++
       drop t (head :: tail)) 0 = _0
    → take (S i) (head :: tail) ++
      drop t (head :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
nth 0 (take (S i) (head :: tail)) 0 = u simpl in *.
dsu: list Slot
head: nat
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (head :: tail)
hV: head = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < S (length tail)
he: match i with
| 0 => head
| S _0 => nth _0 tail 0
end = x
t: nat
hT: match t with
| 0 => head
| S _0 => nth _0 tail 0
end = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < S (length tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (head :: take i tail ++ drop t (head :: tail))
→ ∀ _0 : nat,
    head = _0
    → head
      :: take i tail ++ drop t (head :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
head = u subst head.
dsu: list Slot
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (u :: tail)
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < S (length tail)
he: match i with
| 0 => u
| S _0 => nth _0 tail 0
end = x
t: nat
hT: match t with
| 0 => u
| S _0 => nth _0 tail 0
end = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < S (length tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (u :: take i tail ++ drop t (u :: tail))
→ ∀ _0 : nat,
    u = _0
    → u :: take i tail ++ drop t (u :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
u = u reflexivity. }
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
take (S i) chain ++ drop t chain =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu))
  u
  assert (step2 : validChainToAncestor (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) (take (S i) chain ++ drop t chain)).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (take (S i) chain ++ drop t chain)
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
step2: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
take (S i) chain ++ drop t chain =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu))
  u
  {
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (take (S i) chain ++ drop t chain) repeat split.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
take (S i) chain ++ drop t chain ≠ []
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
nth 0 (take (S i) chain ++ drop t chain) 0 <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
∀ _0 : nat,
  S _0 < length (take (S i) chain ++ drop t chain)
  → nth (nth _0 (take (S i) chain ++ drop t chain) 0)
      (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
         dsu) (Ancestor Unit) =
    ReferTo
      (nth (S _0) (take (S i) chain ++ drop t chain) 0)
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
∃ _0 : Tree,
  nth
    (nth
       (length (take (S i) chain ++ drop t chain) - 1)
       (take (S i) chain ++ drop t chain) 0)
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit) = Ancestor _0
    -
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
take (S i) chain ++ drop t chain ≠ [] destruct chain as [| head tail].
dsu: list Slot
u, x: nat
h0: validChainToAncestor dsu []
hV: nth 0 [] 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length []
he: nth i [] 0 = x
t: nat
hT: nth t [] 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length []
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) [] ++ drop t [])
→ ∀ _0 : nat,
    nth 0 (take (S i) [] ++ drop t []) 0 = _0
    → take (S i) [] ++ drop t [] =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) [] ++ drop t []) 0 = u
take (S i) [] ++ drop t [] ≠ []
dsu: list Slot
head: nat
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (head :: tail)
hV: nth 0 (head :: tail) 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length (head :: tail)
he: nth i (head :: tail) 0 = x
t: nat
hT: nth t (head :: tail) 0 =
ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (head :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (head :: tail) ++
   drop t (head :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (head :: tail) ++
       drop t (head :: tail)) 0 = _0
    → take (S i) (head :: tail) ++
      drop t (head :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0
  (take (S i) (head :: tail) ++
   drop t (head :: tail)) 0 = u
take (S i) (head :: tail) ++ drop t (head :: tail)
≠ [] {
dsu: list Slot
u, x: nat
h0: validChainToAncestor dsu []
hV: nth 0 [] 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length []
he: nth i [] 0 = x
t: nat
hT: nth t [] 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length []
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) [] ++ drop t [])
→ ∀ _0 : nat,
    nth 0 (take (S i) [] ++ drop t []) 0 = _0
    → take (S i) [] ++ drop t [] =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) [] ++ drop t []) 0 = u
take (S i) [] ++ drop t [] ≠ [] cbv in h0.
dsu: list Slot
u, x: nat
h0: (([] = [] → False)
 ∧ 1
   ≤ (fix _0 (_1 : list Slot) : nat :=
        match _1 with
        | [] => 0
        | _ :: _3 => S (_0 _3)
        end) dsu
   ∧ ∀ _0 : nat,
       S (S _0) ≤ 0
       → (fix _1
            (_2 : nat) (_3 : list Slot)
            (_4 : Slot) {struct _3} : Slot :=
            match _2 with
            | 0 =>
                match _3 with
                | [] => _4
                | _5 :: _ => _5
                end
            | S _5 =>
                match _3 with
                | [] => _4
                | _ :: _7 => _1 _5 _7 _4
                end
            end) match _0 with
                 | 0 | _ => 0
                 end dsu 
           (Ancestor Unit) = 
         ReferTo 0)
∧ ∃ _0 : Tree,
    (fix _1
       (_2 : nat) (_3 : list Slot) 
       (_4 : Slot) {struct _3} : Slot :=
       match _2 with
       | 0 =>
           match _3 with
           | [] => _4
           | _5 :: _ => _5
           end
       | S _5 =>
           match _3 with
           | [] => _4
           | _ :: _7 => _1 _5 _7 _4
           end
       end) 0 dsu (Ancestor Unit) = 
    Ancestor _0
hV: nth 0 [] 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length []
he: nth i [] 0 = x
t: nat
hT: nth t [] 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length []
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) [] ++ drop t [])
→ ∀ _0 : nat,
    nth 0 (take (S i) [] ++ drop t []) 0 = _0
    → take (S i) [] ++ drop t [] =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) [] ++ drop t []) 0 = u
take (S i) [] ++ drop t [] ≠ [] easy. }
dsu: list Slot
head: nat
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (head :: tail)
hV: nth 0 (head :: tail) 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length (head :: tail)
he: nth i (head :: tail) 0 = x
t: nat
hT: nth t (head :: tail) 0 =
ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (head :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (head :: tail) ++
   drop t (head :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (head :: tail) ++
       drop t (head :: tail)) 0 = _0
    → take (S i) (head :: tail) ++
      drop t (head :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0
  (take (S i) (head :: tail) ++
   drop t (head :: tail)) 0 = u
take (S i) (head :: tail) ++ drop t (head :: tail)
≠ [] easy.
    -
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
nth 0 (take (S i) chain ++ drop t chain) 0 <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) destruct chain as [| head tail].
dsu: list Slot
u, x: nat
h0: validChainToAncestor dsu []
hV: nth 0 [] 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length []
he: nth i [] 0 = x
t: nat
hT: nth t [] 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length []
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) [] ++ drop t [])
→ ∀ _0 : nat,
    nth 0 (take (S i) [] ++ drop t []) 0 = _0
    → take (S i) [] ++ drop t [] =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) [] ++ drop t []) 0 = u
nth 0 (take (S i) [] ++ drop t []) 0 <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
dsu: list Slot
head: nat
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (head :: tail)
hV: nth 0 (head :: tail) 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length (head :: tail)
he: nth i (head :: tail) 0 = x
t: nat
hT: nth t (head :: tail) 0 =
ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (head :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (head :: tail) ++
   drop t (head :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (head :: tail) ++
       drop t (head :: tail)) 0 = _0
    → take (S i) (head :: tail) ++
      drop t (head :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0
  (take (S i) (head :: tail) ++
   drop t (head :: tail)) 0 = u
nth 0
  (take (S i) (head :: tail) ++ drop t (head :: tail))
  0 <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) {
dsu: list Slot
u, x: nat
h0: validChainToAncestor dsu []
hV: nth 0 [] 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length []
he: nth i [] 0 = x
t: nat
hT: nth t [] 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length []
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) [] ++ drop t [])
→ ∀ _0 : nat,
    nth 0 (take (S i) [] ++ drop t []) 0 = _0
    → take (S i) [] ++ drop t [] =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) [] ++ drop t []) 0 = u
nth 0 (take (S i) [] ++ drop t []) 0 <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) cbv in h0.
dsu: list Slot
u, x: nat
h0: (([] = [] → False)
 ∧ 1
   ≤ (fix _0 (_1 : list Slot) : nat :=
        match _1 with
        | [] => 0
        | _ :: _3 => S (_0 _3)
        end) dsu
   ∧ ∀ _0 : nat,
       S (S _0) ≤ 0
       → (fix _1
            (_2 : nat) (_3 : list Slot)
            (_4 : Slot) {struct _3} : Slot :=
            match _2 with
            | 0 =>
                match _3 with
                | [] => _4
                | _5 :: _ => _5
                end
            | S _5 =>
                match _3 with
                | [] => _4
                | _ :: _7 => _1 _5 _7 _4
                end
            end) match _0 with
                 | 0 | _ => 0
                 end dsu 
           (Ancestor Unit) = 
         ReferTo 0)
∧ ∃ _0 : Tree,
    (fix _1
       (_2 : nat) (_3 : list Slot) 
       (_4 : Slot) {struct _3} : Slot :=
       match _2 with
       | 0 =>
           match _3 with
           | [] => _4
           | _5 :: _ => _5
           end
       | S _5 =>
           match _3 with
           | [] => _4
           | _ :: _7 => _1 _5 _7 _4
           end
       end) 0 dsu (Ancestor Unit) = 
    Ancestor _0
hV: nth 0 [] 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length []
he: nth i [] 0 = x
t: nat
hT: nth t [] 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length []
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) [] ++ drop t [])
→ ∀ _0 : nat,
    nth 0 (take (S i) [] ++ drop t []) 0 = _0
    → take (S i) [] ++ drop t [] =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) [] ++ drop t []) 0 = u
nth 0 (take (S i) [] ++ drop t []) 0 <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) easy. }
dsu: list Slot
head: nat
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (head :: tail)
hV: nth 0 (head :: tail) 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length (head :: tail)
he: nth i (head :: tail) 0 = x
t: nat
hT: nth t (head :: tail) 0 =
ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (head :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (head :: tail) ++
   drop t (head :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (head :: tail) ++
       drop t (head :: tail)) 0 = _0
    → take (S i) (head :: tail) ++
      drop t (head :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0
  (take (S i) (head :: tail) ++
   drop t (head :: tail)) 0 = u
nth 0
  (take (S i) (head :: tail) ++ drop t (head :: tail))
  0 <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) simpl in he.
dsu: list Slot
head: nat
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (head :: tail)
hV: nth 0 (head :: tail) 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length (head :: tail)
he: match i with
| 0 => head
| S _0 => nth _0 tail 0
end = x
t: nat
hT: nth t (head :: tail) 0 =
ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (head :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (head :: tail) ++
   drop t (head :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (head :: tail) ++
       drop t (head :: tail)) 0 = _0
    → take (S i) (head :: tail) ++
      drop t (head :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0
  (take (S i) (head :: tail) ++
   drop t (head :: tail)) 0 = u
nth 0
  (take (S i) (head :: tail) ++ drop t (head :: tail))
  0 <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) rewrite app_nth1; [| rewrite take_length; lia].
dsu: list Slot
head: nat
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (head :: tail)
hV: nth 0 (head :: tail) 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length (head :: tail)
he: match i with
| 0 => head
| S _0 => nth _0 tail 0
end = x
t: nat
hT: nth t (head :: tail) 0 =
ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (head :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (head :: tail) ++
   drop t (head :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (head :: tail) ++
       drop t (head :: tail)) 0 = _0
    → take (S i) (head :: tail) ++
      drop t (head :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0
  (take (S i) (head :: tail) ++
   drop t (head :: tail)) 0 = u
nth 0 (take (S i) (head :: tail)) 0 <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) simpl.
dsu: list Slot
head: nat
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (head :: tail)
hV: nth 0 (head :: tail) 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length (head :: tail)
he: match i with
| 0 => head
| S _0 => nth _0 tail 0
end = x
t: nat
hT: nth t (head :: tail) 0 =
ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (head :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (head :: tail) ++
   drop t (head :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (head :: tail) ++
       drop t (head :: tail)) 0 = _0
    → take (S i) (head :: tail) ++
      drop t (head :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0
  (take (S i) (head :: tail) ++
   drop t (head :: tail)) 0 = u
head <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) simpl in hV.
dsu: list Slot
head: nat
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (head :: tail)
hV: head = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length (head :: tail)
he: match i with
| 0 => head
| S _0 => nth _0 tail 0
end = x
t: nat
hT: nth t (head :: tail) 0 =
ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (head :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (head :: tail) ++
   drop t (head :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (head :: tail) ++
       drop t (head :: tail)) 0 = _0
    → take (S i) (head :: tail) ++
      drop t (head :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0
  (take (S i) (head :: tail) ++
   drop t (head :: tail)) 0 = u
head <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) subst head.
dsu: list Slot
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (u :: tail)
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
he: match i with
| 0 => u
| S _0 => nth _0 tail 0
end = x
hd: i < length (u :: tail)
t: nat
hT: nth t (u :: tail) 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (u :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step1: nth 0
  (take (S i) (u :: tail) ++
   drop t (u :: tail)) 0 = u
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (u :: tail) ++ drop t (u :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (u :: tail) ++
       drop t (u :: tail)) 0 = _0
    → take (S i) (u :: tail) ++
      drop t (u :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
u <
length
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) rewrite insert_length.
dsu: list Slot
tail: list nat
u, x: nat
h0: validChainToAncestor dsu (u :: tail)
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
he: match i with
| 0 => u
| S _0 => nth _0 tail 0
end = x
hd: i < length (u :: tail)
t: nat
hT: nth t (u :: tail) 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length (u :: tail)
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step1: nth 0
  (take (S i) (u :: tail) ++
   drop t (u :: tail)) 0 = u
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (take (S i) (u :: tail) ++ drop t (u :: tail))
→ ∀ _0 : nat,
    nth 0
      (take (S i) (u :: tail) ++
       drop t (u :: tail)) 0 = _0
    → take (S i) (u :: tail) ++
      drop t (u :: tail) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
u < length dsu assumption.
    -
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
∀ _0 : nat,
  S _0 < length (take (S i) chain ++ drop t chain)
  → nth (nth _0 (take (S i) chain ++ drop t chain) 0)
      (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
         dsu) (Ancestor Unit) =
    ReferTo
      (nth (S _0) (take (S i) chain ++ drop t chain) 0) intros a b.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
nth (nth a (take (S i) chain ++ drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S a) (take (S i) chain ++ drop t chain) 0) pose proof h0 as [[e [f g]] [l k]].
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
nth (nth a (take (S i) chain ++ drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S a) (take (S i) chain ++ drop t chain) 0) destruct (ltac:(lia) : a < i \/ a = i \/ i < a) as [hs | [hs | hs]].
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
nth (nth a (take (S i) chain ++ drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S a) (take (S i) chain ++ drop t chain) 0)
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a = i
nth (nth a (take (S i) chain ++ drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S a) (take (S i) chain ++ drop t chain) 0)
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
nth (nth a (take (S i) chain ++ drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S a) (take (S i) chain ++ drop t chain) 0)
      +
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
nth (nth a (take (S i) chain ++ drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S a) (take (S i) chain ++ drop t chain) 0) rewrite !app_nth1; [| rewrite take_length; lia | rewrite take_length; lia].
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
nth (nth a (take (S i) chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo (nth (S a) (take (S i) chain) 0) rewrite <- !nthTake; try lia.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
nth (nth a chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth (S a) chain 0) pose proof list_lookup_insert_ne dsu x (nth a chain 0) (ReferTo (ancestor dsu (length dsu) x)).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
_0: x ≠ nth a chain 0
→ <[x:=ReferTo (ancestor dsu (length dsu) x)]>
    dsu !! nth a chain 0 = 
  dsu !! nth a chain 0
nth (nth a chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth (S a) chain 0) pose proof validChainPairwiseDifferent dsu chain h1 h0 a i ltac:(lia) ltac:(lia) ltac:(lia) as step2.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
_0: x ≠ nth a chain 0
→ <[x:=ReferTo (ancestor dsu (length dsu) x)]>
    dsu !! nth a chain 0 = 
  dsu !! nth a chain 0
step2: nth a chain 0 ≠ nth i chain 0
nth (nth a chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth (S a) chain 0) rewrite he in step2.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
_0: x ≠ nth a chain 0
→ <[x:=ReferTo (ancestor dsu (length dsu) x)]>
    dsu !! nth a chain 0 = 
  dsu !! nth a chain 0
step2: nth a chain 0 ≠ x
nth (nth a chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth (S a) chain 0) pose proof list_lookup_insert_ne dsu x ((nth a chain 0)) (ReferTo (ancestor dsu (length dsu) x)) ltac:(lia) as step3.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
_0: x ≠ nth a chain 0
→ <[x:=ReferTo (ancestor dsu (length dsu) x)]>
    dsu !! nth a chain 0 = 
  dsu !! nth a chain 0
step2: nth a chain 0 ≠ x
step3: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! nth a chain 0 = 
dsu !! nth a chain 0
nth (nth a chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth (S a) chain 0) pose proof (ltac:(clear; intros a b h; subst b; easy) : forall a b, a = b -> default (Ancestor Unit) a = default (Ancestor Unit) b) _ _ step3 as step4.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
_0: x ≠ nth a chain 0
→ <[x:=ReferTo (ancestor dsu (length dsu) x)]>
    dsu !! nth a chain 0 = 
  dsu !! nth a chain 0
step2: nth a chain 0 ≠ x
step3: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! nth a chain 0 = 
dsu !! nth a chain 0
step4: default (Ancestor Unit)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu !! nth a chain 0) =
default (Ancestor Unit) (dsu !! nth a chain 0)
nth (nth a chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth (S a) chain 0) rewrite <- !nth_lookup in step4.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
_0: x ≠ nth a chain 0
→ <[x:=ReferTo (ancestor dsu (length dsu) x)]>
    dsu !! nth a chain 0 = 
  dsu !! nth a chain 0
step2: nth a chain 0 ≠ x
step3: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! nth a chain 0 = 
dsu !! nth a chain 0
step4: nth (nth a chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) =
nth (nth a chain 0) dsu (Ancestor Unit)
nth (nth a chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth (S a) chain 0) rewrite step4.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
_0: x ≠ nth a chain 0
→ <[x:=ReferTo (ancestor dsu (length dsu) x)]>
    dsu !! nth a chain 0 = 
  dsu !! nth a chain 0
step2: nth a chain 0 ≠ x
step3: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! nth a chain 0 = 
dsu !! nth a chain 0
step4: nth (nth a chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) =
nth (nth a chain 0) dsu (Ancestor Unit)
nth (nth a chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S a) chain 0) apply g.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a < i
_0: x ≠ nth a chain 0
→ <[x:=ReferTo (ancestor dsu (length dsu) x)]>
    dsu !! nth a chain 0 = 
  dsu !! nth a chain 0
step2: nth a chain 0 ≠ x
step3: <[x:=ReferTo (ancestor dsu (length dsu) x)]>
  dsu !! nth a chain 0 = 
dsu !! nth a chain 0
step4: nth (nth a chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (Ancestor Unit) =
nth (nth a chain 0) dsu (Ancestor Unit)
S a < length chain lia.
      +
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: a = i
nth (nth a (take (S i) chain ++ drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S a) (take (S i) chain ++ drop t chain) 0) subst a.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
nth (nth i (take (S i) chain ++ drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S i) (take (S i) chain ++ drop t chain) 0) rewrite app_nth1; try (rewrite take_length; lia).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
nth (nth i (take (S i) chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S i) (take (S i) chain ++ drop t chain) 0) rewrite app_nth2; try (rewrite take_length; lia).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
nth (nth i (take (S i) chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S i - length (take (S i) chain))
     (drop t chain) 0) rewrite take_length.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
nth (nth i (take (S i) chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S i - S i `min` length chain) (drop t chain) 0) rewrite (ltac:(lia) : S i - S i `min` length chain = 0).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
nth (nth i (take (S i) chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth 0 (drop t chain) 0) rewrite <- !nthTake; try lia.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
nth (nth i chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth 0 (drop t chain) 0)
        pose proof lookup_drop chain t 0 as step2.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! (t + 0)
nth (nth i chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth 0 (drop t chain) 0) rewrite (ltac:(lia) : t + 0 = t) in step2.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
nth (nth i chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth 0 (drop t chain) 0) pose proof nth_lookup chain t 0 as step3.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (chain !! t)
nth (nth i chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth 0 (drop t chain) 0) rewrite <- step2 in step3.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
nth (nth i chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth 0 (drop t chain) 0)
        pose proof nth_lookup (drop t chain) 0 0 as step4.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
step4: nth 0 (drop t chain) 0 =
default 0 (drop t chain !! 0)
nth (nth i chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth 0 (drop t chain) 0) rewrite <- step3 in step4.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
step4: nth 0 (drop t chain) 0 = nth t chain 0
nth (nth i chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth 0 (drop t chain) 0) rewrite step4.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
step4: nth 0 (drop t chain) 0 = nth t chain 0
nth (nth i chain 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth t chain 0) rewrite he.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
step4: nth 0 (drop t chain) 0 = nth t chain 0
nth x
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = ReferTo (nth t chain 0) rewrite nth_lookup.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
step4: nth 0 (drop t chain) 0 = nth t chain 0
default (Ancestor Unit)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu
   !! x) = ReferTo (nth t chain 0) rewrite list_lookup_insert.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
step4: nth 0 (drop t chain) 0 = nth t chain 0
default (Ancestor Unit)
  (Some (ReferTo (ancestor dsu (length dsu) x))) =
ReferTo (nth t chain 0)
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
step4: nth 0 (drop t chain) 0 = nth t chain 0
x < length dsu
        *
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
step4: nth 0 (drop t chain) 0 = nth t chain 0
default (Ancestor Unit)
  (Some (ReferTo (ancestor dsu (length dsu) x))) =
ReferTo (nth t chain 0) rewrite (ltac:(easy) : forall a b, default a (Some b) = b).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
step4: nth 0 (drop t chain) 0 = nth t chain 0
ReferTo (ancestor dsu (length dsu) x) =
ReferTo (nth t chain 0) rewrite hT.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
step4: nth 0 (drop t chain) 0 = nth t chain 0
ReferTo (ancestor dsu (length dsu) x) =
ReferTo (ancestor dsu (length dsu) x) reflexivity.
        *
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
b: S i < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
step2: drop t chain !! 0 = chain !! t
step3: nth t chain 0 = default 0 (drop t chain !! 0)
step4: nth 0 (drop t chain) 0 = nth t chain 0
x < length dsu lia.
      +
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
nth (nth a (take (S i) chain ++ drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S a) (take (S i) chain ++ drop t chain) 0) rewrite !app_nth2; try (rewrite take_length; lia).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
nth
  (nth (a - length (take (S i) chain)) 
     (drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S a - length (take (S i) chain))
     (drop t chain) 0) rewrite take_length; try lia.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
nth
  (nth (a - S i `min` length chain) (drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo
  (nth (S a - S i `min` length chain) (drop t chain) 0) rewrite (ltac:(lia) : S i `min` length chain = S i).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
nth (nth (a - S i) (drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) =
ReferTo (nth (S a - S i) (drop t chain) 0) rewrite !nth_lookup.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
default (Ancestor Unit)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu
   !! default 0 (drop t chain !! (a - S i))) =
ReferTo (default 0 (drop t chain !! (S a - S i))) rewrite !lookup_drop.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
default (Ancestor Unit)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu
   !! default 0 (chain !! (t + (a - S i)))) =
ReferTo (default 0 (chain !! (t + (S a - S i)))) rewrite (ltac:(lia) : t + (S a - S i) = S (t + (a - S i))).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
default (Ancestor Unit)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu
   !! default 0 (chain !! (t + (a - S i)))) =
ReferTo (default 0 (chain !! S (t + (a - S i)))) rewrite list_lookup_insert_ne; rewrite <- !nth_lookup.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
nth (nth (t + (a - S i)) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (t + (a - S i))) chain 0)
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
x ≠ nth (t + (a - S i)) chain 0
        *
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
nth (nth (t + (a - S i)) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (t + (a - S i))) chain 0) apply g.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
S (t + (a - S i)) < length chain rewrite app_length, take_length, drop_length, (ltac:(lia) : S i `min` length chain = S i) in b.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < S i + (length chain - t)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
S (t + (a - S i)) < length chain lia.
        *
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
x ≠ nth (t + (a - S i)) chain 0 rewrite <- he.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
nth i chain 0 ≠ nth (t + (a - S i)) chain 0 apply (validChainPairwiseDifferent dsu); try (assumption || lia).
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < length (take (S i) chain ++ drop t chain)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
t + (a - S i) < length chain
          rewrite app_length, take_length, drop_length, (ltac:(lia) : S i `min` length chain = S i) in b.
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
a: nat
b: S a < S i + (length chain - t)
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
l: Tree
k: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor l
hs: i < a
t + (a - S i) < length chain lia.
    -
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
∃ _0 : Tree,
  nth
    (nth
       (length (take (S i) chain ++ drop t chain) - 1)
       (take (S i) chain ++ drop t chain) 0)
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit) = Ancestor _0 destruct h0 as [[e [f g]] [b c]].
dsu: list Slot
chain: list nat
u, x: nat
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
∃ _0 : Tree,
  nth
    (nth
       (length (take (S i) chain ++ drop t chain) - 1)
       (take (S i) chain ++ drop t chain) 0)
    (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
    (Ancestor Unit) = Ancestor _0 exists b.
dsu: list Slot
chain: list nat
u, x: nat
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
nth
  (nth (length (take (S i) chain ++ drop t chain) - 1)
     (take (S i) chain ++ drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = Ancestor b rewrite app_length.
dsu: list Slot
chain: list nat
u, x: nat
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
nth
  (nth
     (length (take (S i) chain) +
      length (drop t chain) - 1)
     (take (S i) chain ++ drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = Ancestor b rewrite app_nth2; [| rewrite take_length, drop_length, (ltac:(lia) : S i `min` length chain = S i); lia].
dsu: list Slot
chain: list nat
u, x: nat
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
nth
  (nth
     (length (take (S i) chain) +
      length (drop t chain) - 1 -
      length (take (S i) chain)) 
     (drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = Ancestor b rewrite (ltac:(lia) : length (take (S i) chain) + length (drop t chain) - 1 - length (take (S i) chain) = length (drop t chain) - 1).
dsu: list Slot
chain: list nat
u, x: nat
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
nth (nth (length (drop t chain) - 1) (drop t chain) 0)
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (Ancestor Unit) = Ancestor b rewrite (nth_lookup (drop t chain)), drop_length, lookup_drop, <- nth_lookup, (ltac:(lia) : t + (length chain - t - 1) = length chain - 1), nth_lookup, list_lookup_insert_ne; [now rewrite <- nth_lookup |].
dsu: list Slot
chain: list nat
u, x: nat
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
x ≠ nth (length chain - 1) chain 0 intro gg.
dsu: list Slot
chain: list nat
u, x: nat
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
gg: x = nth (length chain - 1) chain 0
False rewrite <- gg in c.
dsu: list Slot
chain: list nat
u, x: nat
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth x dsu (Ancestor Unit) = Ancestor b
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
gg: x = nth (length chain - 1) chain 0
False rewrite c in h5.
dsu: list Slot
chain: list nat
u, x: nat
e: chain ≠ []
f: nth 0 chain 0 < length dsu
g: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth x dsu (Ancestor Unit) = Ancestor b
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: false
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
gg: x = nth (length chain - 1) chain 0
False easy. }
dsu: list Slot
chain: list nat
u, x: nat
h0: validChainToAncestor dsu chain
hV: nth 0 chain 0 = u
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
i: nat
hd: i < length chain
he: nth i chain 0 = x
t: nat
hT: nth t chain 0 = ancestor dsu (length dsu) x
hT1: i < t
hT2: t < length chain
h1': noIllegalIndices
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
step: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
→ ∀ _0 : nat,
    nth 0 (take (S i) chain ++ drop t chain) 0 =
    _0
    → take (S i) chain ++ drop t chain =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu (length dsu) x)]>
              dsu)) _0
step1: nth 0 (take (S i) chain ++ drop t chain) 0 = u
step2: validChainToAncestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (take (S i) chain ++ drop t chain)
take (S i) chain ++ drop t chain =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu))
  u
      exact (step step2 u step1).
Qed.

Lemma ancestorOfVertexInAncestorChain (dsu : list Slot) chain (u x : nat) (h1 : noIllegalIndices dsu) (h2 : withoutCyclesN dsu (length dsu)) (h3 : u < length dsu) (h4 : x < length dsu) (h5 : match nth x dsu (Ancestor Unit) with | ReferTo _ => true | Ancestor _ => false end) (h6 : validChainToAncestor dsu chain) (h7 : nth 0 chain 0 = u) (s : nat) (h8 : nth s chain 0 = x) (h9 : s < length chain) : ancestor dsu (length dsu) u = ancestor dsu (length dsu) x.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
Proof.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
  rewrite <- !ancestorEqLastAncestorChain, <- (validChainAncestorLength dsu chain h1 h6 _ h7), <- (validChainAncestorLength dsu (drop s chain) h1).
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
nth (length chain - 1) chain 0 =
nth (length (drop s chain) - 1) (drop s chain) 0
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
validChainToAncestor dsu (drop s chain)
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
nth 0 (drop s chain) 0 = x
  -
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
nth (length chain - 1) chain 0 =
nth (length (drop s chain) - 1) (drop s chain) 0 rewrite !nth_lookup, lookup_drop, drop_length, <- !nth_lookup, (ltac:(lia) : s + (length chain - s - 1) = length chain - 1).
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
nth (length chain - 1) chain 0 =
nth (length chain - 1) chain 0 reflexivity.
  -
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
validChainToAncestor dsu (drop s chain) repeat split.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
drop s chain ≠ []
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
nth 0 (drop s chain) 0 < length dsu
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
∀ _0 : nat,
  S _0 < length (drop s chain)
  → nth (nth _0 (drop s chain) 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) (drop s chain) 0)
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
∃ _0 : Tree,
  nth
    (nth (length (drop s chain) - 1) (drop s chain) 0)
    dsu (Ancestor Unit) = 
  Ancestor _0
    +
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
drop s chain ≠ [] intro h.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
h: drop s chain = []
False pose proof (ltac:(rewrite h; reflexivity) : length (drop s chain) = length []) as step1.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
h: drop s chain = []
step1: length (drop s chain) = length []
False rewrite drop_length in step1.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
h: drop s chain = []
step1: length chain - s = length []
False simpl in step1.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
h: drop s chain = []
step1: length chain - s = 0
False lia.
    +
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
nth 0 (drop s chain) 0 < length dsu rewrite nth_lookup, lookup_drop, Nat.add_0_r, <- nth_lookup, h8.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
x < length dsu exact h4.
    +
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
∀ _0 : nat,
  S _0 < length (drop s chain)
  → nth (nth _0 (drop s chain) 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) (drop s chain) 0) intros g i.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
g: nat
i: S g < length (drop s chain)
nth (nth g (drop s chain) 0) dsu (Ancestor Unit) =
ReferTo (nth (S g) (drop s chain) 0) rewrite drop_length in i.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
g: nat
i: S g < length chain - s
nth (nth g (drop s chain) 0) dsu (Ancestor Unit) =
ReferTo (nth (S g) (drop s chain) 0) repeat rewrite (nth_lookup (drop _ _)), lookup_drop.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
g: nat
i: S g < length chain - s
nth (default 0 (chain !! (s + g))) dsu (Ancestor Unit) =
ReferTo (default 0 (chain !! (s + S g))) rewrite <- !nth_lookup, (ltac:(lia) : s + S g = S (s + g)).
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
g: nat
i: S g < length chain - s
nth (nth (s + g) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (s + g)) chain 0)
      destruct h6 as [[e [f j]] [b c]].
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
e: chain ≠ []
f: nth 0 chain 0 < length dsu
j: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
g: nat
i: S g < length chain - s
nth (nth (s + g) chain 0) dsu (Ancestor Unit) =
ReferTo (nth (S (s + g)) chain 0) apply j.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
e: chain ≠ []
f: nth 0 chain 0 < length dsu
j: ∀ _0 : nat,
  S _0 < length chain
  → nth (nth _0 chain 0) dsu (Ancestor Unit) =
    ReferTo (nth (S _0) chain 0)
b: Tree
c: nth (nth (length chain - 1) chain 0) dsu
  (Ancestor Unit) = Ancestor b
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
g: nat
i: S g < length chain - s
S (s + g) < length chain lia.
    +
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
∃ _0 : Tree,
  nth
    (nth (length (drop s chain) - 1) (drop s chain) 0)
    dsu (Ancestor Unit) = 
  Ancestor _0 repeat rewrite (nth_lookup (drop _ _)), lookup_drop.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
∃ _0 : Tree,
  nth
    (default 0
       (chain !! (s + (length (drop s chain) - 1))))
    dsu (Ancestor Unit) = 
  Ancestor _0 rewrite <- !nth_lookup, drop_length, (ltac:(lia) : s + (length chain - s - 1) = length chain - 1).
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
∃ _0 : Tree,
  nth (nth (length chain - 1) chain 0) dsu
    (Ancestor Unit) = Ancestor _0 destruct h6 as [_ c].
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
c: ∃ _0 : Tree,
  nth (nth (length chain - 1) chain 0) dsu
    (Ancestor Unit) = Ancestor _0
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
∃ _0 : Tree,
  nth (nth (length chain - 1) chain 0) dsu
    (Ancestor Unit) = Ancestor _0 exact c.
  -
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
nth 0 (drop s chain) 0 = x rewrite nth_lookup, lookup_drop, Nat.add_0_r, <- nth_lookup.
dsu: list Slot
chain: list nat
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
h6: validChainToAncestor dsu chain
h7: nth 0 chain 0 = u
s: nat
h8: nth s chain 0 = x
h9: s < length chain
nth s chain 0 = x exact h8.
Qed.

Lemma ancestorInsert (dsu : list Slot) (u x : nat) (h1 : noIllegalIndices dsu) (h2 : withoutCyclesN dsu (length dsu)) (h3 : u < length dsu) (h4 : x < length dsu) (h5 : match nth x dsu (Ancestor Unit) with | ReferTo _ => true | Ancestor _ => false end) : ancestor dsu (length dsu) u = ancestor (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu) (length dsu) u.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
Proof.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
  pose proof validChainAncestorChain dsu (length dsu) u h3 h1 as step.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
  pose proof h2 u h3 as step2.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
  remember (existsInRange (length (ancestorChain dsu (length dsu) u)) (fun i => bool_decide (nth i (ancestorChain dsu (length dsu) u) 0 = x))) as s eqn:hs.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: bool
hs: s =
existsInRange
  (length (ancestorChain dsu (length dsu) u))
  (λ _0 : nat,
     bool_decide
       (nth _0 (ancestorChain dsu (length dsu) u)
          0 = x))
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u symmetry in hs.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: bool
hs: existsInRange
  (length (ancestorChain dsu (length dsu) u))
  (λ _0 : nat,
     bool_decide
       (nth _0 (ancestorChain dsu (length dsu) u)
          0 = x)) = s
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u destruct s; [rewrite <- Is_true_true, existsInRangeMeaning in hs | rewrite <- Is_true_false, notExistsInRangeMeaning in hs]; unfold existsInRangeLogic in hs.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∃ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  ∧ bool_decide
      (nth _0 (ancestorChain dsu (length dsu) u)
         0 = x) = true
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: notExistsInRangeLogic
  (length (ancestorChain dsu (length dsu) u))
  (λ _0 : nat,
     bool_decide
       (nth _0 (ancestorChain dsu (length dsu) u)
          0 = x))
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
  -
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∃ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  ∧ bool_decide
      (nth _0 (ancestorChain dsu (length dsu) u)
         0 = x) = true
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u destruct hs as [s [hb hc]].
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: bool_decide
  (nth s (ancestorChain dsu (length dsu) u) 0 = x) =
true
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u rewrite bool_decide_eq_true in hc.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
    assert (step3: validChainToAncestor dsu (ancestorChain dsu (length dsu) u)).
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
    {
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u) split.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
validChain dsu (ancestorChain dsu (length dsu) u)
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) - 1)
       (ancestorChain dsu (length dsu) u) 0) dsu
    (Ancestor Unit) = Ancestor _0 {
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
validChain dsu (ancestorChain dsu (length dsu) u) assumption. }
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) - 1)
       (ancestorChain dsu (length dsu) u) 0) dsu
    (Ancestor Unit) = Ancestor _0 pose proof h2 u h3 as step3.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) - 1)
       (ancestorChain dsu (length dsu) u) 0) dsu
    (Ancestor Unit) = Ancestor _0
      remember (nth (ancestor dsu (length dsu) u) dsu (Ancestor Unit)) as g eqn:hg.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
g: Slot
hg: g =
nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit)
step2: match g with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: match g with
| ReferTo _ => false
| Ancestor _ => true
end
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) - 1)
       (ancestorChain dsu (length dsu) u) 0) dsu
    (Ancestor Unit) = Ancestor _0 symmetry in hg.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
g: Slot
hg: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = g
step2: match g with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: match g with
| ReferTo _ => false
| Ancestor _ => true
end
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) - 1)
       (ancestorChain dsu (length dsu) u) 0) dsu
    (Ancestor Unit) = Ancestor _0 destruct g as [g | g].
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
g: nat
hg: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = ReferTo g
step2: false
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: false
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) - 1)
       (ancestorChain dsu (length dsu) u) 0) dsu
    (Ancestor Unit) = Ancestor _0
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
g: Tree
hg: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = Ancestor g
step2: true
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: true
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) - 1)
       (ancestorChain dsu (length dsu) u) 0) dsu
    (Ancestor Unit) = Ancestor _0 {
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
g: nat
hg: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = ReferTo g
step2: false
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: false
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) - 1)
       (ancestorChain dsu (length dsu) u) 0) dsu
    (Ancestor Unit) = Ancestor _0 exfalso.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
g: nat
hg: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = ReferTo g
step2: false
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: false
False exact step3. }
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
g: Tree
hg: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = Ancestor g
step2: true
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: true
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) - 1)
       (ancestorChain dsu (length dsu) u) 0) dsu
    (Ancestor Unit) = Ancestor _0 exists g.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
g: Tree
hg: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = Ancestor g
step2: true
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: true
nth
  (nth (length (ancestorChain dsu (length dsu) u) - 1)
     (ancestorChain dsu (length dsu) u) 0) dsu
  (Ancestor Unit) = Ancestor g rewrite <- hg, ancestorEqLastAncestorChain.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
g: Tree
hg: nth (ancestor dsu (length dsu) u) dsu
  (Ancestor Unit) = Ancestor g
step2: true
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: true
nth (ancestor dsu (length dsu) u) dsu (Ancestor Unit) =
nth (ancestor dsu (length dsu) u) dsu (Ancestor Unit) reflexivity. }
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
    pose proof ancestorChainInsertPresent dsu (ancestorChain dsu (length dsu) u) u x step3 ltac:(destruct (length dsu); simpl; [reflexivity |]; destruct (nth u dsu (Ancestor Unit)); easy) h1 h2 h3 h4 h5 s hb hc (length (ancestorChain dsu (length dsu) u) - 1) as step4.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: nth
  (length (ancestorChain dsu (length dsu) u) -
   1) (ancestorChain dsu (length dsu) u) 0 =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u rewrite ancestorEqLastAncestorChain in step4.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
    destruct (ltac:(lia) : s = length (ancestorChain dsu (length dsu) u) - 1 \/ s < length (ancestorChain dsu (length dsu) u) - 1) as [step5 | step5].
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
step5: s =
length (ancestorChain dsu (length dsu) u) - 1
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
step5: s <
length (ancestorChain dsu (length dsu) u) - 1
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
    {
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
step5: s =
length (ancestorChain dsu (length dsu) u) - 1
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u subst s.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hc: nth
  (length (ancestorChain dsu (length dsu) u) - 1)
  (ancestorChain dsu (length dsu) u) 0 = x
hb: length (ancestorChain dsu (length dsu) u) - 1 <
length (ancestorChain dsu (length dsu) u)
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ length (ancestorChain dsu (length dsu) u) -
  1 <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take
        (S
           (length
              (ancestorChain dsu 
                 (length dsu) u) - 1))
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u rewrite <- hc in h5.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) -
        1) (ancestorChain dsu (length dsu) u) 0)
    dsu (Ancestor Unit)
with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hc: nth
  (length (ancestorChain dsu (length dsu) u) - 1)
  (ancestorChain dsu (length dsu) u) 0 = x
hb: length (ancestorChain dsu (length dsu) u) - 1 <
length (ancestorChain dsu (length dsu) u)
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ length (ancestorChain dsu (length dsu) u) -
  1 <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take
        (S
           (length
              (ancestorChain dsu 
                 (length dsu) u) - 1))
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u destruct step3 as [[e [f g]] [b c]].
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match
  nth
    (nth
       (length (ancestorChain dsu (length dsu) u) -
        1) (ancestorChain dsu (length dsu) u) 0)
    dsu (Ancestor Unit)
with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hc: nth
  (length (ancestorChain dsu (length dsu) u) - 1)
  (ancestorChain dsu (length dsu) u) 0 = x
hb: length (ancestorChain dsu (length dsu) u) - 1 <
length (ancestorChain dsu (length dsu) u)
e: ancestorChain dsu (length dsu) u ≠ []
f: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
g: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
b: Tree
c: nth
  (nth
     (length (ancestorChain dsu (length dsu) u) -
      1) (ancestorChain dsu (length dsu) u) 0) dsu
  (Ancestor Unit) = Ancestor b
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ length (ancestorChain dsu (length dsu) u) -
  1 <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take
        (S
           (length
              (ancestorChain dsu 
                 (length dsu) u) - 1))
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u rewrite c in h5.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: false
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hc: nth
  (length (ancestorChain dsu (length dsu) u) - 1)
  (ancestorChain dsu (length dsu) u) 0 = x
hb: length (ancestorChain dsu (length dsu) u) - 1 <
length (ancestorChain dsu (length dsu) u)
e: ancestorChain dsu (length dsu) u ≠ []
f: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
g: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
b: Tree
c: nth
  (nth
     (length (ancestorChain dsu (length dsu) u) -
      1) (ancestorChain dsu (length dsu) u) 0) dsu
  (Ancestor Unit) = Ancestor b
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ length (ancestorChain dsu (length dsu) u) -
  1 <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take
        (S
           (length
              (ancestorChain dsu 
                 (length dsu) u) - 1))
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u exfalso.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: false
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hc: nth
  (length (ancestorChain dsu (length dsu) u) - 1)
  (ancestorChain dsu (length dsu) u) 0 = x
hb: length (ancestorChain dsu (length dsu) u) - 1 <
length (ancestorChain dsu (length dsu) u)
e: ancestorChain dsu (length dsu) u ≠ []
f: nth 0 (ancestorChain dsu (length dsu) u) 0 <
length dsu
g: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) u)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) u) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) u) 0)
b: Tree
c: nth
  (nth
     (length (ancestorChain dsu (length dsu) u) -
      1) (ancestorChain dsu (length dsu) u) 0) dsu
  (Ancestor Unit) = Ancestor b
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ length (ancestorChain dsu (length dsu) u) -
  1 <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take
        (S
           (length
              (ancestorChain dsu 
                 (length dsu) u) - 1))
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
False exact h5. }
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
step5: s <
length (ancestorChain dsu (length dsu) u) - 1
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
    pose proof ancestorOfVertexInAncestorChain dsu (ancestorChain dsu (length dsu) u) u x h1 h2 h3 h4 h5 step3 ltac:(destruct (length dsu); simpl; destruct (nth u dsu (Ancestor Unit)); simpl; try lia) s hc hb as step6.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
step5: s <
length (ancestorChain dsu (length dsu) u) - 1
step6: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
    pose proof step4 step6 step5 ltac:(lia) as step7.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
step5: s <
length (ancestorChain dsu (length dsu) u) - 1
step6: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
step7: take (S s) (ancestorChain dsu (length dsu) u) ++
drop
  (length (ancestorChain dsu (length dsu) u) -
   1) (ancestorChain dsu (length dsu) u) =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (length
     (<[x:=ReferTo
             (ancestor dsu (length dsu) x)]>
        dsu)) u
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
    rewrite <- (ancestorEqLastAncestorChain (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)).
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
step5: s <
length (ancestorChain dsu (length dsu) u) - 1
step6: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
step7: take (S s) (ancestorChain dsu (length dsu) u) ++
drop
  (length (ancestorChain dsu (length dsu) u) -
   1) (ancestorChain dsu (length dsu) u) =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu)
  (length
     (<[x:=ReferTo
             (ancestor dsu (length dsu) x)]>
        dsu)) u
ancestor dsu (length dsu) u =
nth
  (length
     (ancestorChain
        (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
           dsu) (length dsu) u) - 1)
  (ancestorChain
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
     (length dsu) u) 0 rewrite insert_length in step7.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
step5: s <
length (ancestorChain dsu (length dsu) u) - 1
step6: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
step7: take (S s) (ancestorChain dsu (length dsu) u) ++
drop
  (length (ancestorChain dsu (length dsu) u) -
   1) (ancestorChain dsu (length dsu) u) =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (length dsu) u
ancestor dsu (length dsu) u =
nth
  (length
     (ancestorChain
        (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
           dsu) (length dsu) u) - 1)
  (ancestorChain
     (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
     (length dsu) u) 0 rewrite <- step7, app_length, app_nth2, drop_length, take_length, (ltac:(lia) : (S s `min` length (ancestorChain dsu (length dsu) u) = S s)); [| rewrite drop_length; lia].
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
step5: s <
length (ancestorChain dsu (length dsu) u) - 1
step6: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
step7: take (S s) (ancestorChain dsu (length dsu) u) ++
drop
  (length (ancestorChain dsu (length dsu) u) -
   1) (ancestorChain dsu (length dsu) u) =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (length dsu) u
ancestor dsu (length dsu) u =
nth
  (S s +
   (length (ancestorChain dsu (length dsu) u) -
    (length (ancestorChain dsu (length dsu) u) - 1)) -
   1 - S s)
  (drop
     (length (ancestorChain dsu (length dsu) u) - 1)
     (ancestorChain dsu (length dsu) u)) 0 rewrite (ltac:(lia) : S s + (length (ancestorChain dsu (length dsu) u) - (length (ancestorChain dsu (length dsu) u) - 1)) - 1 - S s = 0), nth_lookup, lookup_drop, Nat.add_0_r, <- nth_lookup, ancestorEqLastAncestorChain.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
s: nat
hb: s < length (ancestorChain dsu (length dsu) u)
hc: nth s (ancestorChain dsu (length dsu) u) 0 = x
step3: validChainToAncestor dsu
  (ancestorChain dsu (length dsu) u)
step4: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
→ s <
  length (ancestorChain dsu (length dsu) u) -
  1
  → length (ancestorChain dsu (length dsu) u) -
    1 <
    length (ancestorChain dsu (length dsu) u)
    → take (S s)
        (ancestorChain dsu (length dsu) u) ++
      drop
        (length
           (ancestorChain dsu (length dsu) u) -
         1) (ancestorChain dsu (length dsu) u) =
      ancestorChain
        (<[x:=ReferTo
                (ancestor dsu (length dsu) x)]>
           dsu)
        (length
           (<[x:=ReferTo
                   (ancestor dsu 
                      (length dsu) x)]> dsu))
        u
step5: s <
length (ancestorChain dsu (length dsu) u) - 1
step6: ancestor dsu (length dsu) u =
ancestor dsu (length dsu) x
step7: take (S s) (ancestorChain dsu (length dsu) u) ++
drop
  (length (ancestorChain dsu (length dsu) u) -
   1) (ancestorChain dsu (length dsu) u) =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (length dsu) u
ancestor dsu (length dsu) u =
ancestor dsu (length dsu) u reflexivity.
  -
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: notExistsInRangeLogic
  (length (ancestorChain dsu (length dsu) u))
  (λ _0 : nat,
     bool_decide
       (nth _0 (ancestorChain dsu (length dsu) u)
          0 = x))
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u unfold notExistsInRangeLogic in hs.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → ¬ bool_decide
        (nth _0
           (ancestorChain dsu (length dsu) u) 0 =
         x)
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u assert (hd : forall i, i < length (ancestorChain dsu (length dsu) u) -> nth i (ancestorChain dsu (length dsu) u) 0 <> x).
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → ¬ bool_decide
        (nth _0
           (ancestorChain dsu (length dsu) u) 0 =
         x)
∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0 ≠ x
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → ¬ bool_decide
        (nth _0
           (ancestorChain dsu (length dsu) u) 0 =
         x)
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
    {
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → ¬ bool_decide
        (nth _0
           (ancestorChain dsu (length dsu) u) 0 =
         x)
∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0 ≠ x intros a b.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → ¬ bool_decide
        (nth _0
           (ancestorChain dsu (length dsu) u) 0 =
         x)
a: nat
b: a < length (ancestorChain dsu (length dsu) u)
nth a (ancestorChain dsu (length dsu) u) 0 ≠ x pose proof hs a b as c.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → ¬ bool_decide
        (nth _0
           (ancestorChain dsu (length dsu) u) 0 =
         x)
a: nat
b: a < length (ancestorChain dsu (length dsu) u)
c: ¬ bool_decide
    (nth a (ancestorChain dsu (length dsu) u) 0 =
     x)
nth a (ancestorChain dsu (length dsu) u) 0 ≠ x case_bool_decide; [exfalso; exact (c ltac:(easy)) | assumption]. }
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth (ancestor dsu (length dsu) u) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → ¬ bool_decide
        (nth _0
           (ancestorChain dsu (length dsu) u) 0 =
         x)
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u
    rewrite <- (ancestorEqLastAncestorChain dsu (length dsu) u) in step2.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth
    (nth
       (length
          (ancestorChain dsu (length dsu) u) -
        1) (ancestorChain dsu (length dsu) u)
       0) dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → ¬ bool_decide
        (nth _0
           (ancestorChain dsu (length dsu) u) 0 =
         x)
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u pose proof ancestorChainInsertNotPresent dsu (length dsu) u x h1 h2 h3 h4 h5 hd as step3.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth
    (nth
       (length
          (ancestorChain dsu (length dsu) u) -
        1) (ancestorChain dsu (length dsu) u)
       0) dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → ¬ bool_decide
        (nth _0
           (ancestorChain dsu (length dsu) u) 0 =
         x)
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step3: ancestorChain dsu (length dsu) u =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (length dsu) u
ancestor dsu (length dsu) u =
ancestor
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]> dsu)
  (length dsu) u rewrite <- ancestorEqLastAncestorChain, <- ancestorEqLastAncestorChain, <- !step3.
dsu: list Slot
u, x: nat
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
h3: u < length dsu
h4: x < length dsu
h5: match nth x dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
step: validChain dsu
  (ancestorChain dsu (length dsu) u)
step2: match
  nth
    (nth
       (length
          (ancestorChain dsu (length dsu) u) -
        1) (ancestorChain dsu (length dsu) u)
       0) dsu (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
hs: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → ¬ bool_decide
        (nth _0
           (ancestorChain dsu (length dsu) u) 0 =
         x)
hd: ∀ _0 : nat,
  _0 < length (ancestorChain dsu (length dsu) u)
  → nth _0 (ancestorChain dsu (length dsu) u) 0
    ≠ x
step3: ancestorChain dsu (length dsu) u =
ancestorChain
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (length dsu) u
nth (length (ancestorChain dsu (length dsu) u) - 1)
  (ancestorChain dsu (length dsu) u) 0 =
nth (length (ancestorChain dsu (length dsu) u) - 1)
  (ancestorChain dsu (length dsu) u) 0 reflexivity.
Qed.

Fixpoint pathCompress (dsu : list Slot) (fuel : nat) (index ancestor : nat) :=
  match fuel with
  | O => dsu
  | S fuel => match nth index dsu (Ancestor Unit) with
    | ReferTo x => pathCompress (<[index := ReferTo ancestor]> dsu) fuel x ancestor
    | Ancestor _ => dsu
    end
  end.

Lemma pathCompressPreservesNth (dsu : list Slot) a b c d e (hd : nth d dsu (Ancestor Unit) = Ancestor e) : nth d (pathCompress dsu a b c) (Ancestor Unit) = nth d dsu (Ancestor Unit).
dsu: list Slot
a, b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
nth d (pathCompress dsu a b c) (Ancestor Unit) =
nth d dsu (Ancestor Unit)
Proof.
dsu: list Slot
a, b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
nth d (pathCompress dsu a b c) (Ancestor Unit) =
nth d dsu (Ancestor Unit)
  revert dsu b c d e hd.
a: nat
∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2) (Ancestor Unit) =
    nth _3 _0 (Ancestor Unit) induction a as [| a IH].
∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 0 _1 _2) (Ancestor Unit) =
    nth _3 _0 (Ancestor Unit)
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 (S a) _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit) {
∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 0 _1 _2) (Ancestor Unit) =
    nth _3 _0 (Ancestor Unit) easy. }
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 (S a) _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
  intros dsu b c d e hd.
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
nth d (pathCompress dsu (S a) b c) (Ancestor Unit) =
nth d dsu (Ancestor Unit) simpl.
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
nth d
  match nth b dsu (Ancestor Unit) with
  | ReferTo _0 =>
      pathCompress (<[b:=ReferTo c]> dsu) a _0 c
  | Ancestor _ => dsu
  end (Ancestor Unit) = nth d dsu (Ancestor Unit) remember (nth b dsu (Ancestor Unit)) as x eqn:hx.
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: Slot
hx: x = nth b dsu (Ancestor Unit)
nth d
  match x with
  | ReferTo _0 =>
      pathCompress (<[b:=ReferTo c]> dsu) a _0 c
  | Ancestor _ => dsu
  end (Ancestor Unit) = nth d dsu (Ancestor Unit) destruct x as [x | x]; symmetry in hx; [| reflexivity].
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
nth d (pathCompress (<[b:=ReferTo c]> dsu) a x c)
  (Ancestor Unit) = nth d dsu (Ancestor Unit) pose proof IH (<[b:=ReferTo c]> dsu) x c d e as step.
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit)
nth d (pathCompress (<[b:=ReferTo c]> dsu) a x c)
  (Ancestor Unit) = nth d dsu (Ancestor Unit) destruct (decide (b = d)) as [hs | hs].
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit)
hs: b = d
nth d (pathCompress (<[b:=ReferTo c]> dsu) a x c)
  (Ancestor Unit) = nth d dsu (Ancestor Unit)
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit)
hs: b ≠ d
nth d (pathCompress (<[b:=ReferTo c]> dsu) a x c)
  (Ancestor Unit) = nth d dsu (Ancestor Unit) {
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit)
hs: b = d
nth d (pathCompress (<[b:=ReferTo c]> dsu) a x c)
  (Ancestor Unit) = nth d dsu (Ancestor Unit) subst b.
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
step: nth d (<[d:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[d:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[d:=ReferTo c]> dsu) (Ancestor Unit)
hx: nth d dsu (Ancestor Unit) = ReferTo x
nth d (pathCompress (<[d:=ReferTo c]> dsu) a x c)
  (Ancestor Unit) = nth d dsu (Ancestor Unit) rewrite hx in hd.
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
c, d: nat
e: Tree
x: nat
hd: ReferTo x = Ancestor e
step: nth d (<[d:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[d:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[d:=ReferTo c]> dsu) (Ancestor Unit)
hx: nth d dsu (Ancestor Unit) = ReferTo x
nth d (pathCompress (<[d:=ReferTo c]> dsu) a x c)
  (Ancestor Unit) = nth d dsu (Ancestor Unit) easy. }
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit)
hs: b ≠ d
nth d (pathCompress (<[b:=ReferTo c]> dsu) a x c)
  (Ancestor Unit) = nth d dsu (Ancestor Unit)
  assert (step2: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) = nth d dsu (Ancestor Unit)).
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit)
hs: b ≠ d
nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
nth d dsu (Ancestor Unit)
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit)
hs: b ≠ d
step2: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
nth d dsu (Ancestor Unit)
nth d (pathCompress (<[b:=ReferTo c]> dsu) a x c)
  (Ancestor Unit) = nth d dsu (Ancestor Unit)
  {
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit)
hs: b ≠ d
nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
nth d dsu (Ancestor Unit) rewrite nth_lookup, list_lookup_insert_ne; [| exact hs].
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit)
hs: b ≠ d
default (Ancestor Unit) (dsu !! d) =
nth d dsu (Ancestor Unit) rewrite <- nth_lookup.
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit)
hs: b ≠ d
nth d dsu (Ancestor Unit) = nth d dsu (Ancestor Unit) reflexivity. }
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) =
  nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit)
hs: b ≠ d
step2: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
nth d dsu (Ancestor Unit)
nth d (pathCompress (<[b:=ReferTo c]> dsu) a x c)
  (Ancestor Unit) = nth d dsu (Ancestor Unit) rewrite step2 in step.
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d dsu (Ancestor Unit) = Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) = nth d dsu (Ancestor Unit)
hs: b ≠ d
step2: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
nth d dsu (Ancestor Unit)
nth d (pathCompress (<[b:=ReferTo c]> dsu) a x c)
  (Ancestor Unit) = nth d dsu (Ancestor Unit) rewrite (step hd).
a: nat
IH: ∀ (_0 : list Slot) (_1 _2 _3 : nat) (_4 : Tree),
  nth _3 _0 (Ancestor Unit) = Ancestor _4
  → nth _3 (pathCompress _0 a _1 _2)
      (Ancestor Unit) = nth _3 _0 (Ancestor Unit)
dsu: list Slot
b, c, d: nat
e: Tree
hd: nth d dsu (Ancestor Unit) = Ancestor e
x: nat
hx: nth b dsu (Ancestor Unit) = ReferTo x
step: nth d dsu (Ancestor Unit) = Ancestor e
→ nth d
    (pathCompress (<[b:=ReferTo c]> dsu) a x c)
    (Ancestor Unit) = nth d dsu (Ancestor Unit)
hs: b ≠ d
step2: nth d (<[b:=ReferTo c]> dsu) (Ancestor Unit) =
nth d dsu (Ancestor Unit)
nth d dsu (Ancestor Unit) = nth d dsu (Ancestor Unit) reflexivity.
Qed.

Lemma withoutCyclesExtractAncestor (dsu : list Slot) (h2 : withoutCyclesN dsu (length dsu)) (a : nat) (hA : a < length dsu) : exists k, nth (ancestor dsu (length dsu) a) dsu (Ancestor Unit) = Ancestor k.
dsu: list Slot
h2: withoutCyclesN dsu (length dsu)
a: nat
hA: a < length dsu
∃ _0 : Tree,
  nth (ancestor dsu (length dsu) a) dsu
    (Ancestor Unit) = Ancestor _0
Proof.
dsu: list Slot
h2: withoutCyclesN dsu (length dsu)
a: nat
hA: a < length dsu
∃ _0 : Tree,
  nth (ancestor dsu (length dsu) a) dsu
    (Ancestor Unit) = Ancestor _0
  pose proof h2 a hA as step.
dsu: list Slot
h2: withoutCyclesN dsu (length dsu)
a: nat
hA: a < length dsu
step: match
  nth (ancestor dsu (length dsu) a) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
∃ _0 : Tree,
  nth (ancestor dsu (length dsu) a) dsu
    (Ancestor Unit) = Ancestor _0
  destruct (nth (ancestor dsu (length dsu) a) dsu (Ancestor Unit)) as [g | g].
dsu: list Slot
h2: withoutCyclesN dsu (length dsu)
a: nat
hA: a < length dsu
g: nat
step: false
∃ _0 : Tree, ReferTo g = Ancestor _0
dsu: list Slot
h2: withoutCyclesN dsu (length dsu)
a: nat
hA: a < length dsu
g: Tree
step: true
∃ _0 : Tree, Ancestor g = Ancestor _0 {
dsu: list Slot
h2: withoutCyclesN dsu (length dsu)
a: nat
hA: a < length dsu
g: nat
step: false
∃ _0 : Tree, ReferTo g = Ancestor _0 exfalso.
dsu: list Slot
h2: withoutCyclesN dsu (length dsu)
a: nat
hA: a < length dsu
g: nat
step: false
False exact step. }
dsu: list Slot
h2: withoutCyclesN dsu (length dsu)
a: nat
hA: a < length dsu
g: Tree
step: true
∃ _0 : Tree, Ancestor g = Ancestor _0 exists g.
dsu: list Slot
h2: withoutCyclesN dsu (length dsu)
a: nat
hA: a < length dsu
g: Tree
step: true
Ancestor g = Ancestor g reflexivity.
Qed.

Lemma pathCompressNth (dsu : list Slot) (h1 : noIllegalIndices dsu) (fuel : nat) (a b c : nat) (hA : a < length dsu) (hB : b < length dsu) (hC : c < length dsu) : nth c (pathCompress dsu fuel a b) (Ancestor Unit) = nth c dsu (Ancestor Unit) \/ nth c (pathCompress dsu fuel a b) (Ancestor Unit) = ReferTo b.
dsu: list Slot
h1: noIllegalIndices dsu
fuel, a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
nth c (pathCompress dsu fuel a b) (Ancestor Unit) =
nth c dsu (Ancestor Unit)
∨ nth c (pathCompress dsu fuel a b) (Ancestor Unit) =
  ReferTo b
Proof.
dsu: list Slot
h1: noIllegalIndices dsu
fuel, a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
nth c (pathCompress dsu fuel a b) (Ancestor Unit) =
nth c dsu (Ancestor Unit)
∨ nth c (pathCompress dsu fuel a b) (Ancestor Unit) =
  ReferTo b
  revert dsu h1 a b c hA hB hC.
fuel: nat
∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2 induction fuel as [| fuel IH];
  intros dsu h1 a b c hA hB hC.
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
nth c (pathCompress dsu 0 a b) (Ancestor Unit) =
nth c dsu (Ancestor Unit)
∨ nth c (pathCompress dsu 0 a b) (Ancestor Unit) =
  ReferTo b
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
nth c (pathCompress dsu (S fuel) a b) (Ancestor Unit) =
nth c dsu (Ancestor Unit)
∨ nth c (pathCompress dsu (S fuel) a b)
    (Ancestor Unit) = ReferTo b
  {
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
nth c (pathCompress dsu 0 a b) (Ancestor Unit) =
nth c dsu (Ancestor Unit)
∨ nth c (pathCompress dsu 0 a b) (Ancestor Unit) =
  ReferTo b simpl.
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
nth c dsu (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c dsu (Ancestor Unit) = ReferTo b left.
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
nth c dsu (Ancestor Unit) = nth c dsu (Ancestor Unit) reflexivity. }
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
nth c (pathCompress dsu (S fuel) a b) (Ancestor Unit) =
nth c dsu (Ancestor Unit)
∨ nth c (pathCompress dsu (S fuel) a b)
    (Ancestor Unit) = ReferTo b
  simpl.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
nth c
  match nth a dsu (Ancestor Unit) with
  | ReferTo _0 =>
      pathCompress (<[a:=ReferTo b]> dsu) fuel _0 b
  | Ancestor _ => dsu
  end (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c
    match nth a dsu (Ancestor Unit) with
    | ReferTo _0 =>
        pathCompress (<[a:=ReferTo b]> dsu) fuel _0 b
    | Ancestor _ => dsu
    end (Ancestor Unit) = 
  ReferTo b remember (nth a dsu (Ancestor Unit)) as e eqn:he.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: Slot
he: e = nth a dsu (Ancestor Unit)
nth c
  match e with
  | ReferTo _0 =>
      pathCompress (<[a:=ReferTo b]> dsu) fuel _0 b
  | Ancestor _ => dsu
  end (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c
    match e with
    | ReferTo _0 =>
        pathCompress (<[a:=ReferTo b]> dsu) fuel _0 b
    | Ancestor _ => dsu
    end (Ancestor Unit) = 
  ReferTo b symmetry in he.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: Slot
he: nth a dsu (Ancestor Unit) = e
nth c
  match e with
  | ReferTo _0 =>
      pathCompress (<[a:=ReferTo b]> dsu) fuel _0 b
  | Ancestor _ => dsu
  end (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c
    match e with
    | ReferTo _0 =>
        pathCompress (<[a:=ReferTo b]> dsu) fuel _0 b
    | Ancestor _ => dsu
    end (Ancestor Unit) = 
  ReferTo b
  destruct e as [e | e]; [| left; reflexivity].
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
  pose proof h1 a _ he as hE.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
  epose proof IH (<[a:=ReferTo b]> dsu) _ e b c ltac:(rewrite insert_length; exact hE) ltac:(rewrite insert_length; exact hB) ltac:(rewrite insert_length; exact hC) as hf.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
hf: nth c
  (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) =
nth c (<[a:=ReferTo b]> dsu) (Ancestor Unit)
∨ nth c
    (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
  destruct (decide (c = a)) as [hs | hs].
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
hf: nth c
  (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) =
nth c (<[a:=ReferTo b]> dsu) (Ancestor Unit)
∨ nth c
    (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
hs: c = a
nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
hf: nth c
  (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) =
nth c (<[a:=ReferTo b]> dsu) (Ancestor Unit)
∨ nth c
    (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
hs: c ≠ a
nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
  -
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
hf: nth c
  (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) =
nth c (<[a:=ReferTo b]> dsu) (Ancestor Unit)
∨ nth c
    (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
hs: c = a
nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b subst c.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b: nat
hA: a < length dsu
hB: b < length dsu
hC: a < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
hf: nth a
  (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) =
nth a (<[a:=ReferTo b]> dsu) (Ancestor Unit)
∨ nth a
    (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
nth a (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = nth a dsu (Ancestor Unit)
∨ nth a (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b rewrite (ltac:(rewrite nth_lookup, list_lookup_insert; easy) : nth a (<[a:=ReferTo b]> dsu) (Ancestor Unit) = ReferTo b) in hf.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b: nat
hA: a < length dsu
hB: b < length dsu
hC: a < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
hf: nth a
  (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = ReferTo b
∨ nth a
    (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
nth a (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = nth a dsu (Ancestor Unit)
∨ nth a (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b right.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b: nat
hA: a < length dsu
hB: b < length dsu
hC: a < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
hf: nth a
  (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = ReferTo b
∨ nth a
    (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
nth a (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = ReferTo b destruct hf; assumption.
  -
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
hf: nth c
  (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) =
nth c (<[a:=ReferTo b]> dsu) (Ancestor Unit)
∨ nth c
    (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
hs: c ≠ a
nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b rewrite (ltac:(rewrite !nth_lookup, list_lookup_insert_ne; try lia; reflexivity) : nth c (<[a:=ReferTo b]> dsu) (Ancestor Unit) = nth c dsu (Ancestor Unit)) in hf.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
hf: nth c
  (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c
    (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b
hs: c ≠ a
nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
  (Ancestor Unit) = nth c dsu (Ancestor Unit)
∨ nth c (pathCompress (<[a:=ReferTo b]> dsu) fuel e b)
    (Ancestor Unit) = ReferTo b exact hf.
  Unshelve.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
noIllegalIndices (<[a:=ReferTo b]> dsu)
  intros i j hi.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i (<[a:=ReferTo b]> dsu) (Ancestor Unit) =
ReferTo j
j < length (<[a:=ReferTo b]> dsu) rewrite insert_length.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i (<[a:=ReferTo b]> dsu) (Ancestor Unit) =
ReferTo j
j < length dsu destruct (decide (i = a)) as [hs | hs].
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i (<[a:=ReferTo b]> dsu) (Ancestor Unit) =
ReferTo j
hs: i = a
j < length dsu
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i (<[a:=ReferTo b]> dsu) (Ancestor Unit) =
ReferTo j
hs: i ≠ a
j < length dsu
  +
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i (<[a:=ReferTo b]> dsu) (Ancestor Unit) =
ReferTo j
hs: i = a
j < length dsu subst i.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
j: nat
hi: nth a (<[a:=ReferTo b]> dsu) (Ancestor Unit) =
ReferTo j
j < length dsu rewrite (ltac:(rewrite nth_lookup, list_lookup_insert; easy) : nth a (<[a:=ReferTo b]> dsu) (Ancestor Unit) = ReferTo b) in hi.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
j: nat
hi: ReferTo b = ReferTo j
j < length dsu injection hi.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
j: nat
hi: ReferTo b = ReferTo j
b = j → j < length dsu lia.
  +
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i (<[a:=ReferTo b]> dsu) (Ancestor Unit) =
ReferTo j
hs: i ≠ a
j < length dsu rewrite (ltac:(rewrite !nth_lookup, list_lookup_insert_ne; try lia; reflexivity) : nth i (<[a:=ReferTo b]> dsu) (Ancestor Unit) = nth i dsu (Ancestor Unit)) in hi.
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i dsu (Ancestor Unit) = ReferTo j
hs: i ≠ a
j < length dsu
    assert (hi1 : i < length dsu).
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i dsu (Ancestor Unit) = ReferTo j
hs: i ≠ a
i < length dsu
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i dsu (Ancestor Unit) = ReferTo j
hs: i ≠ a
hi1: i < length dsu
j < length dsu
    {
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i dsu (Ancestor Unit) = ReferTo j
hs: i ≠ a
i < length dsu destruct (decide (length dsu <= i)) as [ht | ht]; [| lia].
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i dsu (Ancestor Unit) = ReferTo j
hs: i ≠ a
ht: length dsu ≤ i
i < length dsu rewrite nth_overflow in hi; [| exact ht].
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: Ancestor Unit = ReferTo j
hs: i ≠ a
ht: length dsu ≤ i
i < length dsu easy. }
fuel: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → ∀ _1 _2 _3 : nat,
      _1 < length _0
      → _2 < length _0
        → _3 < length _0
          → nth _3 (pathCompress _0 fuel _1 _2)
              (Ancestor Unit) =
            nth _3 _0 (Ancestor Unit)
            ∨ nth _3 (pathCompress _0 fuel _1 _2)
                (Ancestor Unit) = 
              ReferTo _2
dsu: list Slot
h1: noIllegalIndices dsu
a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
e: nat
he: nth a dsu (Ancestor Unit) = ReferTo e
hE: e < length dsu
i, j: nat
hi: nth i dsu (Ancestor Unit) = ReferTo j
hs: i ≠ a
hi1: i < length dsu
j < length dsu
    exact (h1 i _ hi).
Qed.

Lemma doubleBangEqIfNthEq {A} (l l1 : list A) (hL : length l = length l1) d (hnth : forall x, x < length l -> nth x l d = nth x l1 d) n : l !! n = l1 !! n.
A: Type
l, l1: list A
hL: length l = length l1
d: A
hnth: ∀ _0 : nat,
  _0 < length l → nth _0 l d = nth _0 l1 d
n: nat
l !! n = l1 !! n
Proof.
A: Type
l, l1: list A
hL: length l = length l1
d: A
hnth: ∀ _0 : nat,
  _0 < length l → nth _0 l d = nth _0 l1 d
n: nat
l !! n = l1 !! n
  remember (l !! n) as target1 eqn:h1.
A: Type
l, l1: list A
hL: length l = length l1
d: A
hnth: ∀ _0 : nat,
  _0 < length l → nth _0 l d = nth _0 l1 d
n: nat
target1: option A
h1: target1 = l !! n
target1 = l1 !! n
  remember (l1 !! n) as target2 eqn:h2.
A: Type
l, l1: list A
hL: length l = length l1
d: A
hnth: ∀ _0 : nat,
  _0 < length l → nth _0 l d = nth _0 l1 d
n: nat
target1: option A
h1: target1 = l !! n
target2: option A
h2: target2 = l1 !! n
target1 = target2
  symmetry in h1.
A: Type
l, l1: list A
hL: length l = length l1
d: A
hnth: ∀ _0 : nat,
  _0 < length l → nth _0 l d = nth _0 l1 d
n: nat
target1: option A
h1: l !! n = target1
target2: option A
h2: target2 = l1 !! n
target1 = target2 symmetry in h2.
A: Type
l, l1: list A
hL: length l = length l1
d: A
hnth: ∀ _0 : nat,
  _0 < length l → nth _0 l d = nth _0 l1 d
n: nat
target1: option A
h1: l !! n = target1
target2: option A
h2: l1 !! n = target2
target1 = target2
  destruct target1 as [target1 |]; destruct target2 as [target2 |]; try fast_reflexivity; try pose proof lookup_ge_None_1 _ _ h1 as d1; try pose proof lookup_ge_None_1 _ _ h2 as d2; try pose proof lookup_ge_None_2 _ _ (ltac:(lia) : length l <= n); try pose proof lookup_ge_None_2 _ _ (ltac:(lia) : length l1 <= n); try congruence.
A: Type
l, l1: list A
hL: length l = length l1
d: A
hnth: ∀ _0 : nat,
  _0 < length l → nth _0 l d = nth _0 l1 d
n: nat
target1: A
h1: l !! n = Some target1
target2: A
h2: l1 !! n = Some target2
Some target1 = Some target2
  pose proof lookup_lt_Some _ _ _ h1 as s.
A: Type
l, l1: list A
hL: length l = length l1
d: A
hnth: ∀ _0 : nat,
  _0 < length l → nth _0 l d = nth _0 l1 d
n: nat
target1: A
h1: l !! n = Some target1
target2: A
h2: l1 !! n = Some target2
s: n < length l
Some target1 = Some target2
  pose proof hnth n s as t.
A: Type
l, l1: list A
hL: length l = length l1
d: A
hnth: ∀ _0 : nat,
  _0 < length l → nth _0 l d = nth _0 l1 d
n: nat
target1: A
h1: l !! n = Some target1
target2: A
h2: l1 !! n = Some target2
s: n < length l
t: nth n l d = nth n l1 d
Some target1 = Some target2
  rewrite !nth_lookup, h1, h2 in t.
A: Type
l, l1: list A
hL: length l = length l1
d: A
hnth: ∀ _0 : nat,
  _0 < length l → nth _0 l d = nth _0 l1 d
n: nat
target1: A
h1: l !! n = Some target1
target2: A
h2: l1 !! n = Some target2
s: n < length l
t: default d (Some target1) =
default d (Some target2)
Some target1 = Some target2 simpl in t.
A: Type
l, l1: list A
hL: length l = length l1
d: A
hnth: ∀ _0 : nat,
  _0 < length l → nth _0 l d = nth _0 l1 d
n: nat
target1: A
h1: l !! n = Some target1
target2: A
h2: l1 !! n = Some target2
s: n < length l
t: target1 = target2
Some target1 = Some target2 congruence.
Qed.

Definition performMerge (dsu : list Slot) (tree1 tree2 : Tree) (u v : nat) :=
  <[u := ReferTo v]> (<[v := Ancestor (Unite tree2 tree1)]> dsu).

Lemma performMergeScore' (dsu : list Slot) (tree1 tree2 : Tree) (u v : nat) (hUV : u < v) (hL : v < length dsu) (hU : nth u dsu (Ancestor Unit) = Ancestor tree1) (hV : nth v dsu (Ancestor Unit) = Ancestor tree2) : (dsuScore (performMerge dsu tree1 tree2 u v) = dsuScore dsu + Z.of_nat (leafCount tree1) + Z.of_nat (leafCount tree2))%Z.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore (performMerge dsu tree1 tree2 u v) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
Proof.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore (performMerge dsu tree1 tree2 u v) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
  unfold performMerge.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore
  (<[u:=ReferTo v]>
     (<[v:=Ancestor (Unite tree2 tree1)]> dsu)) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite (insert_take_drop dsu); [| exact hL].
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore
  (<[u:=ReferTo v]>
     (take v dsu ++
      Ancestor (Unite tree2 tree1) :: drop (S v) dsu)) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite insert_app_l; [| rewrite take_length; lia].
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore
  (<[u:=ReferTo v]> (take v dsu) ++
   Ancestor (Unite tree2 tree1) :: drop (S v) dsu) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite insert_take_drop, take_take; [| rewrite take_length; lia].
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore
  ((take (u `min` v) dsu ++
    ReferTo v :: drop (S u) (take v dsu)) ++
   Ancestor (Unite tree2 tree1) :: drop (S v) dsu) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite (ltac:(lia) : u `min` v = u).
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore
  ((take u dsu ++ ReferTo v :: drop (S u) (take v dsu)) ++
   Ancestor (Unite tree2 tree1) :: drop (S v) dsu) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
  pose proof ListDecomposition.listDecomposition dsu u v hUV hL (Ancestor Unit) as a.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
a: dsu =
take u dsu ++
[nth u dsu (Ancestor Unit)] ++
drop (S u) (take v dsu) ++
[nth v dsu (Ancestor Unit)] ++ drop (v + 1) dsu
dsuScore
  ((take u dsu ++ ReferTo v :: drop (S u) (take v dsu)) ++
   Ancestor (Unite tree2 tree1) :: drop (S v) dsu) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite a at 4.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
a: dsu =
take u dsu ++
[nth u dsu (Ancestor Unit)] ++
drop (S u) (take v dsu) ++
[nth v dsu (Ancestor Unit)] ++ drop (v + 1) dsu
dsuScore
  ((take u dsu ++ ReferTo v :: drop (S u) (take v dsu)) ++
   Ancestor (Unite tree2 tree1) :: drop (S v) dsu) =
(dsuScore
   (take u dsu ++
    [nth u dsu (Ancestor Unit)] ++
    drop (S u) (take v dsu) ++
    [nth v dsu (Ancestor Unit)] ++ drop (v + 1) dsu) +
 Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite hU, hV, ((ltac:(intros; easy) : forall a b, a :: b = [a] ++ b) (ReferTo _)), ((ltac:(intros; easy) : forall a b, a :: b = [a] ++ b) (Ancestor _)), <- !app_assoc.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
a: dsu =
take u dsu ++
[nth u dsu (Ancestor Unit)] ++
drop (S u) (take v dsu) ++
[nth v dsu (Ancestor Unit)] ++ drop (v + 1) dsu
dsuScore
  (take u dsu ++
   [ReferTo v] ++
   drop (S u) (take v dsu) ++
   [Ancestor (Unite tree2 tree1)] ++ drop (S v) dsu) =
(dsuScore
   (take u dsu ++
    [Ancestor tree1] ++
    drop (S u) (take v dsu) ++
    [Ancestor tree2] ++ drop (v + 1) dsu) +
 Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
  unfold dsuScore.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
a: dsu =
take u dsu ++
[nth u dsu (Ancestor Unit)] ++
drop (S u) (take v dsu) ++
[nth v dsu (Ancestor Unit)] ++ drop (v + 1) dsu
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end)
        (take u dsu ++
         [ReferTo v] ++
         drop (S u) (take v dsu) ++
         [Ancestor (Unite tree2 tree1)] ++
         drop (S v) dsu))) =
(Z.of_nat
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0%nat
            | Ancestor _1 => score _1
            end)
         (take u dsu ++
          [Ancestor tree1] ++
          drop (S u) (take v dsu) ++
          [Ancestor tree2] ++ drop (v + 1) dsu))) +
 Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite !map_app, !list_sum_app.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
a: dsu =
take u dsu ++
[nth u dsu (Ancestor Unit)] ++
drop (S u) (take v dsu) ++
[nth v dsu (Ancestor Unit)] ++ drop (v + 1) dsu
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) (take u dsu)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) [ReferTo v]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => score _1
             end) (drop (S u) (take v dsu))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => score _1
              end) [Ancestor (Unite tree2 tree1)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => score _1
              end) (drop (S v) dsu)))))) =
(Z.of_nat
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0%nat
            | Ancestor _1 => score _1
            end) (take u dsu)) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0%nat
             | Ancestor _1 => score _1
             end) [Ancestor tree1]) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0%nat
              | Ancestor _1 => score _1
              end) (drop (S u) (take v dsu))) +
      (list_sum
         (map
            (λ _0 : Slot,
               match _0 with
               | ReferTo _ => 0%nat
               | Ancestor _1 => score _1
               end) [Ancestor tree2]) +
       list_sum
         (map
            (λ _0 : Slot,
               match _0 with
               | ReferTo _ => 0%nat
               | Ancestor _1 => score _1
               end) (drop (v + 1) dsu)))))) +
 Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z simpl.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
a: dsu =
take u dsu ++
[nth u dsu (Ancestor Unit)] ++
drop (S u) (take v dsu) ++
[nth v dsu (Ancestor Unit)] ++ drop (v + 1) dsu
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) (take u dsu)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) (drop (S u) (take v dsu))) +
    (leafCount tree2 + leafCount tree1 + score tree2 +
     score tree1 + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => score _1
             end) (drop (S v) dsu))))) =
(Z.of_nat
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0%nat
            | Ancestor _1 => score _1
            end) (take u dsu)) +
    (score tree1 + 0 +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0%nat
              | Ancestor _1 => score _1
              end) (drop (S u) (take v dsu))) +
      (score tree2 + 0 +
       list_sum
         (map
            (λ _0 : Slot,
               match _0 with
               | ReferTo _ => 0%nat
               | Ancestor _1 => score _1
               end) (drop (v + 1) dsu)))))) +
 Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite (ltac:(easy) : _ + 1 = S _), !Nat.add_0_r.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u < v
hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
a: dsu =
take u dsu ++
[nth u dsu (Ancestor Unit)] ++
drop (S u) (take v dsu) ++
[nth v dsu (Ancestor Unit)] ++ drop (v + 1) dsu
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) (take u dsu)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) (drop (S u) (take v dsu))) +
    (leafCount tree2 + leafCount tree1 + score tree2 +
     score tree1 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => score _1
             end) (drop (S v) dsu))))) =
(Z.of_nat
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0%nat
            | Ancestor _1 => score _1
            end) (take u dsu)) +
    (score tree1 +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0%nat
              | Ancestor _1 => score _1
              end) (drop (S u) (take v dsu))) +
      (score tree2 +
       list_sum
         (map
            (λ _0 : Slot,
               match _0 with
               | ReferTo _ => 0%nat
               | Ancestor _1 => score _1
               end) (drop (S v) dsu)))))) +
 Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z lia.
Qed.

Lemma performMergeScore'' (dsu : list Slot) (tree1 tree2 : Tree) (u v : nat) (hUV : v < u) (hL : u < length dsu) (hU : nth u dsu (Ancestor Unit) = Ancestor tree1) (hV : nth v dsu (Ancestor Unit) = Ancestor tree2) : (dsuScore (performMerge dsu tree1 tree2 u v) = dsuScore dsu + Z.of_nat (leafCount tree1) + Z.of_nat (leafCount tree2))%Z.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore (performMerge dsu tree1 tree2 u v) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
Proof.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore (performMerge dsu tree1 tree2 u v) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
  unfold performMerge.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore
  (<[u:=ReferTo v]>
     (<[v:=Ancestor (Unite tree2 tree1)]> dsu)) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite (insert_take_drop dsu); [| ltac:(lia)].
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore
  (<[u:=ReferTo v]>
     (take v dsu ++
      Ancestor (Unite tree2 tree1) :: drop (S v) dsu)) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
  assert (h : forall a (b : list Slot), <[u:=a]> b =  <[length (take v dsu) + (u - v):=a]> b).
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
h: ∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
dsuScore
  (<[u:=ReferTo v]>
     (take v dsu ++
      Ancestor (Unite tree2 tree1) :: drop (S v) dsu)) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
  {
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1 intros a b.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
a: Slot
b: list Slot
<[u:=a]> b = <[length (take v dsu) + (u - v):=a]> b rewrite take_length, (ltac:(lia) : v `min` length dsu + (u - v) = u).
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
a: Slot
b: list Slot
<[u:=a]> b = <[u:=a]> b reflexivity. }
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
h: ∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
dsuScore
  (<[u:=ReferTo v]>
     (take v dsu ++
      Ancestor (Unite tree2 tree1) :: drop (S v) dsu)) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite h.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
h: ∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
dsuScore
  (<[length (take v dsu) + (u - v):=
     ReferTo v]>
     (take v dsu ++
      Ancestor (Unite tree2 tree1) :: drop (S v) dsu)) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
  rewrite insert_app_r, ((ltac:(intros; easy) : forall a b, a :: b = [a] ++ b) (Ancestor _)), (ltac:(simpl; lia) : u - v = length [Ancestor (Unite tree2 tree1)] + (u - v - 1)), insert_app_r.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
h: ∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
dsuScore
  (take v dsu ++
   [Ancestor (Unite tree2 tree1)] ++
   <[u - v - 1:=ReferTo v]> (drop (S v) dsu)) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite insert_take_drop, drop_drop; [| rewrite drop_length; lia].
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
h: ∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
dsuScore
  (take v dsu ++
   [Ancestor (Unite tree2 tree1)] ++
   take (u - v - 1) (drop (S v) dsu) ++
   ReferTo v :: drop (S v + S (u - v - 1)) dsu) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite (ltac:(lia) : S v + S (u - v - 1) = u + 1), (ltac:(easy) : ReferTo v :: drop (u + 1) dsu = [ReferTo v] ++ drop (u + 1) dsu).
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
h: ∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
dsuScore
  (take v dsu ++
   [Ancestor (Unite tree2 tree1)] ++
   take (u - v - 1) (drop (S v) dsu) ++
   [ReferTo v] ++ drop (u + 1) dsu) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z unfold dsuScore.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
h: ∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end)
        (take v dsu ++
         [Ancestor (Unite tree2 tree1)] ++
         take (u - v - 1) (drop (S v) dsu) ++
         [ReferTo v] ++ drop (u + 1) dsu))) =
(Z.of_nat
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0%nat
            | Ancestor _1 => score _1
            end) dsu)) + 
 Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
  pose proof ListDecomposition.listDecomposition dsu v u hUV hL (Ancestor Unit) as a.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
h: ∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
a: dsu =
take v dsu ++
[nth v dsu (Ancestor Unit)] ++
drop (S v) (take u dsu) ++
[nth u dsu (Ancestor Unit)] ++ drop (u + 1) dsu
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end)
        (take v dsu ++
         [Ancestor (Unite tree2 tree1)] ++
         take (u - v - 1) (drop (S v) dsu) ++
         [ReferTo v] ++ drop (u + 1) dsu))) =
(Z.of_nat
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0%nat
            | Ancestor _1 => score _1
            end) dsu)) + 
 Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite a at 4.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
h: ∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
a: dsu =
take v dsu ++
[nth v dsu (Ancestor Unit)] ++
drop (S v) (take u dsu) ++
[nth u dsu (Ancestor Unit)] ++ drop (u + 1) dsu
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end)
        (take v dsu ++
         [Ancestor (Unite tree2 tree1)] ++
         take (u - v - 1) (drop (S v) dsu) ++
         [ReferTo v] ++ drop (u + 1) dsu))) =
(Z.of_nat
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0%nat
            | Ancestor _1 => score _1
            end)
         (take v dsu ++
          [nth v dsu (Ancestor Unit)] ++
          drop (S v) (take u dsu) ++
          [nth u dsu (Ancestor Unit)] ++
          drop (u + 1) dsu))) +
 Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z rewrite take_drop_commute, (ltac:(lia) : S v + (u - v - 1) = u), !map_app, !list_sum_app, hU, hV.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
h: ∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
a: dsu =
take v dsu ++
[nth v dsu (Ancestor Unit)] ++
drop (S v) (take u dsu) ++
[nth u dsu (Ancestor Unit)] ++ drop (u + 1) dsu
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) (take v dsu)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) [Ancestor (Unite tree2 tree1)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => score _1
             end) (drop (S v) (take u dsu))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => score _1
              end) [ReferTo v]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => score _1
              end) (drop (u + 1) dsu)))))) =
(Z.of_nat
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0%nat
            | Ancestor _1 => score _1
            end) (take v dsu)) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0%nat
             | Ancestor _1 => score _1
             end) [Ancestor tree2]) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0%nat
              | Ancestor _1 => score _1
              end) (drop (S v) (take u dsu))) +
      (list_sum
         (map
            (λ _0 : Slot,
               match _0 with
               | ReferTo _ => 0%nat
               | Ancestor _1 => score _1
               end) [Ancestor tree1]) +
       list_sum
         (map
            (λ _0 : Slot,
               match _0 with
               | ReferTo _ => 0%nat
               | Ancestor _1 => score _1
               end) (drop (u + 1) dsu)))))) +
 Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z simpl.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: v < u
hL: u < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
h: ∀ (_0 : Slot) (_1 : list Slot),
  <[u:=_0]> _1 =
  <[length (take v dsu) + (u - v):=_0]> _1
a: dsu =
take v dsu ++
[nth v dsu (Ancestor Unit)] ++
drop (S v) (take u dsu) ++
[nth u dsu (Ancestor Unit)] ++ drop (u + 1) dsu
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) (take v dsu)) +
   (leafCount tree2 + leafCount tree1 + score tree2 +
    score tree1 + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => score _1
             end) (drop (S v) (take u dsu))) +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => score _1
             end) (drop (u + 1) dsu))))) =
(Z.of_nat
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0%nat
            | Ancestor _1 => score _1
            end) (take v dsu)) +
    (score tree2 + 0 +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0%nat
              | Ancestor _1 => score _1
              end) (drop (S v) (take u dsu))) +
      (score tree1 + 0 +
       list_sum
         (map
            (λ _0 : Slot,
               match _0 with
               | ReferTo _ => 0%nat
               | Ancestor _1 => score _1
               end) (drop (u + 1) dsu)))))) +
 Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z lia.
Qed.

Lemma performMergeScore (dsu : list Slot) (tree1 tree2 : Tree) (u v : nat) (hUV : u <> v) (hLt1 : u < length dsu) (hLt2 : v < length dsu) (hL : v < length dsu) (hU : nth u dsu (Ancestor Unit) = Ancestor tree1) (hV : nth v dsu (Ancestor Unit) = Ancestor tree2) : (dsuScore (performMerge dsu tree1 tree2 u v) = dsuScore dsu + Z.of_nat (leafCount tree1) + Z.of_nat (leafCount tree2))%Z.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u ≠ v
hLt1: u < length dsu
hLt2, hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore (performMerge dsu tree1 tree2 u v) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
Proof.
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u ≠ v
hLt1: u < length dsu
hLt2, hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
dsuScore (performMerge dsu tree1 tree2 u v) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
  destruct (ltac:(lia) : u < v \/ v < u) as [hs | hs].
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u ≠ v
hLt1: u < length dsu
hLt2, hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
hs: u < v
dsuScore (performMerge dsu tree1 tree2 u v) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u ≠ v
hLt1: u < length dsu
hLt2, hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
hs: v < u
dsuScore (performMerge dsu tree1 tree2 u v) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z
  -
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u ≠ v
hLt1: u < length dsu
hLt2, hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
hs: u < v
dsuScore (performMerge dsu tree1 tree2 u v) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z apply performMergeScore'; assumption.
  -
dsu: list Slot
tree1, tree2: Tree
u, v: nat
hUV: u ≠ v
hLt1: u < length dsu
hLt2, hL: v < length dsu
hU: nth u dsu (Ancestor Unit) = Ancestor tree1
hV: nth v dsu (Ancestor Unit) = Ancestor tree2
hs: v < u
dsuScore (performMerge dsu tree1 tree2 u v) =
(dsuScore dsu + Z.of_nat (leafCount tree1) +
 Z.of_nat (leafCount tree2))%Z apply performMergeScore''; assumption.
Qed.

Definition unite (dsu : list Slot) (a b : nat) :=
  if decide (length dsu <= a) then dsu else
  if decide (length dsu <= b) then dsu else
  let u := ancestor dsu (length dsu) a in
  let dsu1 := pathCompress dsu (length dsu) a u in
  let v := ancestor dsu1 (length dsu1) b in
  let dsu2 := pathCompress dsu1 (length dsu1) b v in
  if decide (u = v) then dsu2 else
  match nth u dsu2 (Ancestor Unit) with
  | ReferTo _ => dsu2
  | Ancestor tree1 => match nth v dsu2 (Ancestor Unit) with
    | ReferTo _ => dsu2
    | Ancestor tree2 =>
      if decide (leafCount tree2 < leafCount tree1) then performMerge dsu2 tree2 tree1 v u else performMerge dsu2 tree1 tree2 u v
    end
  end.

Fixpoint dsuFromInteractions (dsu : list Slot) (interactions : list (nat * nat)) :=
  match interactions with
  | [] => dsu
  | (a, b)::tail => if decide (a < length dsu /\ b < length dsu) then dsuFromInteractions (unite dsu a b) tail else dsuFromInteractions dsu tail
  end.

Lemma pathCompressPreservesLength (dsu : list Slot) (n a b : nat) : length (pathCompress dsu n a b) = length dsu.
dsu: list Slot
n, a, b: nat
length (pathCompress dsu n a b) = length dsu
Proof.
dsu: list Slot
n, a, b: nat
length (pathCompress dsu n a b) = length dsu
  induction n as [| n IH] in dsu, a |- *; [easy |].
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  length (pathCompress _0 n _1 b) = length _0
length (pathCompress dsu (S n) a b) = length dsu
  simpl.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  length (pathCompress _0 n _1 b) = length _0
length
  match nth a dsu (Ancestor Unit) with
  | ReferTo _0 =>
      pathCompress (<[a:=ReferTo b]> dsu) n _0 b
  | Ancestor _ => dsu
  end = length dsu destruct (nth a dsu (Ancestor Unit)) as [x | x]; [| easy].
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  length (pathCompress _0 n _1 b) = length _0
x: nat
length (pathCompress (<[a:=ReferTo b]> dsu) n x b) =
length dsu rewrite IH.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  length (pathCompress _0 n _1 b) = length _0
x: nat
length (<[a:=ReferTo b]> dsu) = length dsu apply insert_length.
Qed.

Lemma pathCompressPreservesScore (dsu : list Slot) (n a b : nat) : dsuScore (pathCompress dsu n a b) = dsuScore dsu.
dsu: list Slot
n, a, b: nat
dsuScore (pathCompress dsu n a b) = dsuScore dsu
Proof.
dsu: list Slot
n, a, b: nat
dsuScore (pathCompress dsu n a b) = dsuScore dsu
  induction n as [| n IH] in dsu, a |- *; [easy |].
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
dsuScore (pathCompress dsu (S n) a b) = dsuScore dsu
  simpl.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
dsuScore
  match nth a dsu (Ancestor Unit) with
  | ReferTo _0 =>
      pathCompress (<[a:=ReferTo b]> dsu) n _0 b
  | Ancestor _ => dsu
  end = dsuScore dsu
  remember (nth a dsu (Ancestor Unit)) as x eqn:hx.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: Slot
hx: x = nth a dsu (Ancestor Unit)
dsuScore
  match x with
  | ReferTo _0 =>
      pathCompress (<[a:=ReferTo b]> dsu) n _0 b
  | Ancestor _ => dsu
  end = dsuScore dsu symmetry in hx.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: Slot
hx: nth a dsu (Ancestor Unit) = x
dsuScore
  match x with
  | ReferTo _0 =>
      pathCompress (<[a:=ReferTo b]> dsu) n _0 b
  | Ancestor _ => dsu
  end = dsuScore dsu destruct x as [x | x]; [| reflexivity].
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
dsuScore (pathCompress (<[a:=ReferTo b]> dsu) n x b) =
dsuScore dsu
  rewrite IH.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
dsuScore (<[a:=ReferTo b]> dsu) = dsuScore dsu
  destruct (decide (a < length dsu)) as [ha | ha]; [| rewrite list_insert_ge; lia].
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
dsuScore (<[a:=ReferTo b]> dsu) = dsuScore dsu
  rewrite insert_take_drop; [| exact ha].
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
dsuScore (take a dsu ++ ReferTo b :: drop (S a) dsu) =
dsuScore dsu unfold dsuScore.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end)
        (take a dsu ++ ReferTo b :: drop (S a) dsu))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) dsu)) rewrite (ltac:(intros; listsEqual) : forall a b, a :: b = [a] ++ b), !map_app, !list_sum_app.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) (take a dsu)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) [ReferTo b]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) (drop (S a) dsu)))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) dsu))
  assert (s : list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) [ReferTo b]) = list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) [nth a dsu (Ancestor Unit)])).
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo b]) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [nth a dsu (Ancestor Unit)])
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
s: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo b]) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [nth a dsu (Ancestor Unit)])
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) (take a dsu)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) [ReferTo b]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) (drop (S a) dsu)))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) dsu))
  {
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo b]) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [nth a dsu (Ancestor Unit)]) rewrite hx.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo b]) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo x]) easy. }
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
s: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo b]) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [nth a dsu (Ancestor Unit)])
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) (take a dsu)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) [ReferTo b]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => score _1
            end) (drop (S a) dsu)))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) dsu))
  rewrite s, <- !list_sum_app, <- !map_app, <- ListDecomposition.listDecompositionSingle.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
s: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo b]) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [nth a dsu (Ancestor Unit)])
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) dsu)) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) dsu))
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
s: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo b]) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [nth a dsu (Ancestor Unit)])
a < length dsu {
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
s: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo b]) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [nth a dsu (Ancestor Unit)])
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) dsu)) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) dsu)) reflexivity. }
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuScore (pathCompress _0 n _1 b) = dsuScore _0
x: nat
hx: nth a dsu (Ancestor Unit) = ReferTo x
ha: a < length dsu
s: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo b]) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [nth a dsu (Ancestor Unit)])
a < length dsu exact ha.
Qed.

Lemma pathCompressPreservesLeafCount (dsu : list Slot) (n a b : nat) : dsuLeafCount (pathCompress dsu n a b) = dsuLeafCount dsu.
dsu: list Slot
n, a, b: nat
dsuLeafCount (pathCompress dsu n a b) =
dsuLeafCount dsu
Proof.
dsu: list Slot
n, a, b: nat
dsuLeafCount (pathCompress dsu n a b) =
dsuLeafCount dsu
  induction n as [| n IH] in dsu, a |- *.
dsu: list Slot
a, b: nat
dsuLeafCount (pathCompress dsu 0 a b) =
dsuLeafCount dsu
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuLeafCount (pathCompress _0 n _1 b) =
  dsuLeafCount _0
dsuLeafCount (pathCompress dsu (S n) a b) =
dsuLeafCount dsu {
dsu: list Slot
a, b: nat
dsuLeafCount (pathCompress dsu 0 a b) =
dsuLeafCount dsu easy. }
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuLeafCount (pathCompress _0 n _1 b) =
  dsuLeafCount _0
dsuLeafCount (pathCompress dsu (S n) a b) =
dsuLeafCount dsu
  simpl.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuLeafCount (pathCompress _0 n _1 b) =
  dsuLeafCount _0
dsuLeafCount
  match nth a dsu (Ancestor Unit) with
  | ReferTo _0 =>
      pathCompress (<[a:=ReferTo b]> dsu) n _0 b
  | Ancestor _ => dsu
  end = dsuLeafCount dsu remember (nth a dsu (Ancestor Unit)) as x eqn:hX.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuLeafCount (pathCompress _0 n _1 b) =
  dsuLeafCount _0
x: Slot
hX: x = nth a dsu (Ancestor Unit)
dsuLeafCount
  match x with
  | ReferTo _0 =>
      pathCompress (<[a:=ReferTo b]> dsu) n _0 b
  | Ancestor _ => dsu
  end = dsuLeafCount dsu symmetry in hX.
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuLeafCount (pathCompress _0 n _1 b) =
  dsuLeafCount _0
x: Slot
hX: nth a dsu (Ancestor Unit) = x
dsuLeafCount
  match x with
  | ReferTo _0 =>
      pathCompress (<[a:=ReferTo b]> dsu) n _0 b
  | Ancestor _ => dsu
  end = dsuLeafCount dsu destruct x as [x | x]; [| reflexivity].
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuLeafCount (pathCompress _0 n _1 b) =
  dsuLeafCount _0
x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
dsuLeafCount
  (pathCompress (<[a:=ReferTo b]> dsu) n x b) =
dsuLeafCount dsu
  rewrite (IH (<[a := ReferTo b]> dsu) x).
dsu: list Slot
n, a, b: nat
IH: ∀ (_0 : list Slot) (_1 : nat),
  dsuLeafCount (pathCompress _0 n _1 b) =
  dsuLeafCount _0
x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
dsuLeafCount (<[a:=ReferTo b]> dsu) = dsuLeafCount dsu clear IH n.
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
dsuLeafCount (<[a:=ReferTo b]> dsu) = dsuLeafCount dsu destruct (decide (length dsu <= a)) as [hL | hL].
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: length dsu ≤ a
dsuLeafCount (<[a:=ReferTo b]> dsu) = dsuLeafCount dsu
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
dsuLeafCount (<[a:=ReferTo b]> dsu) = dsuLeafCount dsu {
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: length dsu ≤ a
dsuLeafCount (<[a:=ReferTo b]> dsu) = dsuLeafCount dsu rewrite list_insert_ge; lia. }
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
dsuLeafCount (<[a:=ReferTo b]> dsu) = dsuLeafCount dsu rewrite insert_take_drop; [| lia].
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
dsuLeafCount
  (take a dsu ++ ReferTo b :: drop (S a) dsu) =
dsuLeafCount dsu unfold dsuLeafCount.
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end)
        (take a dsu ++ ReferTo b :: drop (S a) dsu))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) dsu)) apply inj_eq.
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end)
     (take a dsu ++ ReferTo b :: drop (S a) dsu)) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) dsu) rewrite map_app, list_sum_app.
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) (take a dsu)) +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) (ReferTo b :: drop (S a) dsu)) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) dsu) simpl.
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) (take a dsu)) +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) (drop (S a) dsu)) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) dsu) rewrite (ltac:(lia) : forall a b, a + b = a + 0 + b).
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) (take a dsu)) + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) (drop (S a) dsu)) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) dsu) rewrite (ltac:(easy) : 0 = list_sum (map (fun x => match x with | ReferTo x => 0 | Ancestor x => leafCount x end) [ReferTo x])) at 1.
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) (take a dsu)) +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) [ReferTo x]) +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) (drop (S a) dsu)) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) dsu) rewrite <- hX, <- !list_sum_app, <- !map_app.
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end)
     ((take a dsu ++ [nth a dsu (Ancestor Unit)]) ++
      drop (S a) dsu)) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) dsu)
  assert (H1 : forall (T : Type) (l : list T) i default (H : i < length l), take i l ++ [nth i l default] = take (S i) l).
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
∀ (_0 : Type) (_1 : list _0) (_2 : nat) (_3 : _0),
  _2 < length _1
  → take _2 _1 ++ [nth _2 _1 _3] = take (S _2) _1
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
H1: ∀ (_0 : Type) (_1 : list _0) (_2 : nat) (_3 : _0),
  _2 < length _1
  → take _2 _1 ++ [nth _2 _1 _3] = take (S _2) _1
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end)
     ((take a dsu ++ [nth a dsu (Ancestor Unit)]) ++
      drop (S a) dsu)) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) dsu)
  {
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
∀ (_0 : Type) (_1 : list _0) (_2 : nat) (_3 : _0),
  _2 < length _1
  → take _2 _1 ++ [nth _2 _1 _3] = take (S _2) _1 clear.
∀ (_0 : Type) (_1 : list _0) (_2 : nat) (_3 : _0),
  _2 < length _1
  → take _2 _1 ++ [nth _2 _1 _3] = take (S _2) _1 intros T l i default H.
T: Type
l: list T
i: nat
default: T
H: i < length l
take i l ++ [nth i l default] = take (S i) l
    revert H.
T: Type
l: list T
i: nat
default: T
i < length l
→ take i l ++ [nth i l default] = take (S i) l revert l.
T: Type
i: nat
default: T
∀ _0 : list T,
  i < length _0
  → take i _0 ++ [nth i _0 default] = take (S i) _0 induction i as [| i IHi]; intros l H.
T: Type
default: T
l: list T
H: 0 < length l
take 0 l ++ [nth 0 l default] = take 1 l
T: Type
i: nat
default: T
IHi: ∀ _0 : list T,
  i < length _0
  → take i _0 ++ [nth i _0 default] =
    take (S i) _0
l: list T
H: S i < length l
take (S i) l ++ [nth (S i) l default] =
take (S (S i)) l
    -
T: Type
default: T
l: list T
H: 0 < length l
take 0 l ++ [nth 0 l default] = take 1 l rewrite take_0.
T: Type
default: T
l: list T
H: 0 < length l
[] ++ [nth 0 l default] = take 1 l destruct l; simpl in *; try lia.
T: Type
default, _0: T
_1: list T
H: 0 < S (length _1)
[_0] = _0 :: take 0 _1 rewrite take_0.
T: Type
default, _0: T
_1: list T
H: 0 < S (length _1)
[_0] = [_0] reflexivity.
    -
T: Type
i: nat
default: T
IHi: ∀ _0 : list T,
  i < length _0
  → take i _0 ++ [nth i _0 default] =
    take (S i) _0
l: list T
H: S i < length l
take (S i) l ++ [nth (S i) l default] =
take (S (S i)) l destruct l; simpl in H; try lia.
T: Type
i: nat
default: T
IHi: ∀ _0 : list T,
  i < length _0
  → take i _0 ++ [nth i _0 default] =
    take (S i) _0
_0: T
_1: list T
H: S i < S (length _1)
take (S i) (_0 :: _1) ++
[nth (S i) (_0 :: _1) default] =
take (S (S i)) (_0 :: _1)
      simpl.
T: Type
i: nat
default: T
IHi: ∀ _0 : list T,
  i < length _0
  → take i _0 ++ [nth i _0 default] =
    take (S i) _0
_0: T
_1: list T
H: S i < S (length _1)
_0 :: take i _1 ++ [nth i _1 default] =
_0 :: take (S i) _1 rewrite IHi; try lia.
T: Type
i: nat
default: T
IHi: ∀ _0 : list T,
  i < length _0
  → take i _0 ++ [nth i _0 default] =
    take (S i) _0
_0: T
_1: list T
H: S i < S (length _1)
_0 :: take (S i) _1 = _0 :: take (S i) _1 reflexivity. }
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
H1: ∀ (_0 : Type) (_1 : list _0) (_2 : nat) (_3 : _0),
  _2 < length _1
  → take _2 _1 ++ [nth _2 _1 _3] = take (S _2) _1
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end)
     ((take a dsu ++ [nth a dsu (Ancestor Unit)]) ++
      drop (S a) dsu)) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) dsu)
  rewrite H1; [| lia].
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
H1: ∀ (_0 : Type) (_1 : list _0) (_2 : nat) (_3 : _0),
  _2 < length _1
  → take _2 _1 ++ [nth _2 _1 _3] = take (S _2) _1
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) (take (S a) dsu ++ drop (S a) dsu)) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) dsu) rewrite take_drop.
dsu: list Slot
a, b, x: nat
hX: nth a dsu (Ancestor Unit) = ReferTo x
hL: ¬ length dsu ≤ a
H1: ∀ (_0 : Type) (_1 : list _0) (_2 : nat) (_3 : _0),
  _2 < length _1
  → take _2 _1 ++ [nth _2 _1 _3] = take (S _2) _1
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) dsu) =
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) dsu) reflexivity.
Qed.

Lemma pathCompressPreservesNoIllegalIndices (dsu : list Slot) (h1 : noIllegalIndices dsu) (n u v : nat) (hU : u < length dsu) (hV : v < length dsu) : noIllegalIndices (pathCompress dsu n u v).
dsu: list Slot
h1: noIllegalIndices dsu
n, u, v: nat
hU: u < length dsu
hV: v < length dsu
noIllegalIndices (pathCompress dsu n u v)
Proof.
dsu: list Slot
h1: noIllegalIndices dsu
n, u, v: nat
hU: u < length dsu
hV: v < length dsu
noIllegalIndices (pathCompress dsu n u v)
  revert u hU.
dsu: list Slot
h1: noIllegalIndices dsu
n, v: nat
hV: v < length dsu
∀ _0 : nat,
  _0 < length dsu
  → noIllegalIndices (pathCompress dsu n _0 v) revert h1.
dsu: list Slot
n, v: nat
hV: v < length dsu
noIllegalIndices dsu
→ ∀ _0 : nat,
    _0 < length dsu
    → noIllegalIndices (pathCompress dsu n _0 v) revert hV.
dsu: list Slot
n, v: nat
v < length dsu
→ noIllegalIndices dsu
  → ∀ _0 : nat,
      _0 < length dsu
      → noIllegalIndices (pathCompress dsu n _0 v) revert dsu.
n, v: nat
∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices (pathCompress _0 n _1 v) induction n as [| n IH]; intros dsu hV h1 u hU.
v: nat
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
noIllegalIndices (pathCompress dsu 0 u v)
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
noIllegalIndices (pathCompress dsu (S n) u v)
  -
v: nat
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
noIllegalIndices (pathCompress dsu 0 u v) simpl.
v: nat
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
noIllegalIndices dsu intros.
v: nat
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
noIllegalIndices dsu exact h1.
  -
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
noIllegalIndices (pathCompress dsu (S n) u v) simpl.
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
noIllegalIndices
  match nth u dsu (Ancestor Unit) with
  | ReferTo _0 =>
      pathCompress (<[u:=ReferTo v]> dsu) n _0 v
  | Ancestor _ => dsu
  end intros i j hi.
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j: nat
hi: nth i
  match nth u dsu (Ancestor Unit) with
  | ReferTo _0 =>
      pathCompress (<[u:=ReferTo v]> dsu) n _0 v
  | Ancestor _ => dsu
  end (Ancestor Unit) = ReferTo j
j <
length
  match nth u dsu (Ancestor Unit) with
  | ReferTo _0 =>
      pathCompress (<[u:=ReferTo v]> dsu) n _0 v
  | Ancestor _ => dsu
  end remember (nth u dsu (Ancestor Unit)) as e eqn:he.
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j: nat
e: Slot
he: e = nth u dsu (Ancestor Unit)
hi: nth i
  match e with
  | ReferTo _0 =>
      pathCompress (<[u:=ReferTo v]> dsu) n _0 v
  | Ancestor _ => dsu
  end (Ancestor Unit) = ReferTo j
j <
length
  match e with
  | ReferTo _0 =>
      pathCompress (<[u:=ReferTo v]> dsu) n _0 v
  | Ancestor _ => dsu
  end symmetry in he.
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j: nat
e: Slot
he: nth u dsu (Ancestor Unit) = e
hi: nth i
  match e with
  | ReferTo _0 =>
      pathCompress (<[u:=ReferTo v]> dsu) n _0 v
  | Ancestor _ => dsu
  end (Ancestor Unit) = ReferTo j
j <
length
  match e with
  | ReferTo _0 =>
      pathCompress (<[u:=ReferTo v]> dsu) n _0 v
  | Ancestor _ => dsu
  end destruct e as [e | e].
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
j < length (pathCompress (<[u:=ReferTo v]> dsu) n e v)
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j: nat
e: Tree
he: nth u dsu (Ancestor Unit) = Ancestor e
hi: nth i dsu (Ancestor Unit) = ReferTo j
j < length dsu
    +
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
j < length (pathCompress (<[u:=ReferTo v]> dsu) n e v) assert (h : noIllegalIndices (<[u:=ReferTo v]> dsu)).
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
noIllegalIndices (<[u:=ReferTo v]> dsu)
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
h: noIllegalIndices (<[u:=ReferTo v]> dsu)
j < length (pathCompress (<[u:=ReferTo v]> dsu) n e v)
      {
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
noIllegalIndices (<[u:=ReferTo v]> dsu) intros a b c.
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
a, b: nat
c: nth a (<[u:=ReferTo v]> dsu) (Ancestor Unit) =
ReferTo b
b < length (<[u:=ReferTo v]> dsu) destruct (decide (a = u)) as [ha | ha].
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
a, b: nat
c: nth a (<[u:=ReferTo v]> dsu) (Ancestor Unit) =
ReferTo b
ha: a = u
b < length (<[u:=ReferTo v]> dsu)
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
a, b: nat
c: nth a (<[u:=ReferTo v]> dsu) (Ancestor Unit) =
ReferTo b
ha: a ≠ u
b < length (<[u:=ReferTo v]> dsu)
        -
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
a, b: nat
c: nth a (<[u:=ReferTo v]> dsu) (Ancestor Unit) =
ReferTo b
ha: a = u
b < length (<[u:=ReferTo v]> dsu) rewrite ha, nth_lookup, list_lookup_insert, (ltac:(intros; easy) : forall a, default _ (Some a) = a) in c; [| exact hU].
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
a, b: nat
c: ReferTo v = ReferTo b
ha: a = u
b < length (<[u:=ReferTo v]> dsu) injection c.
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
a, b: nat
c: ReferTo v = ReferTo b
ha: a = u
v = b → b < length (<[u:=ReferTo v]> dsu) intro d.
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
a, b: nat
c: ReferTo v = ReferTo b
ha: a = u
d: v = b
b < length (<[u:=ReferTo v]> dsu) subst v.
n, b: nat
c: ReferTo b = ReferTo b
IH: ∀ _0 : list Slot,
  b < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 b)
dsu: list Slot
hV: b < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo b]> dsu) n e b)
  (Ancestor Unit) = ReferTo j
a: nat
ha: a = u
b < length (<[u:=ReferTo b]> dsu) rewrite insert_length.
n, b: nat
c: ReferTo b = ReferTo b
IH: ∀ _0 : list Slot,
  b < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 b)
dsu: list Slot
hV: b < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo b]> dsu) n e b)
  (Ancestor Unit) = ReferTo j
a: nat
ha: a = u
b < length dsu exact hV.
        -
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
a, b: nat
c: nth a (<[u:=ReferTo v]> dsu) (Ancestor Unit) =
ReferTo b
ha: a ≠ u
b < length (<[u:=ReferTo v]> dsu) rewrite insert_length.
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
a, b: nat
c: nth a (<[u:=ReferTo v]> dsu) (Ancestor Unit) =
ReferTo b
ha: a ≠ u
b < length dsu rewrite nth_lookup, list_lookup_insert_ne in c; [| lia].
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
a, b: nat
c: default (Ancestor Unit) (dsu !! a) = ReferTo b
ha: a ≠ u
b < length dsu rewrite <- (nth_lookup _ _ (Ancestor Unit)) in c.
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
a, b: nat
c: nth a dsu (Ancestor Unit) = ReferTo b
ha: a ≠ u
b < length dsu exact (h1 _ _ c). }
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j, e: nat
he: nth u dsu (Ancestor Unit) = ReferTo e
hi: nth i (pathCompress (<[u:=ReferTo v]> dsu) n e v)
  (Ancestor Unit) = ReferTo j
h: noIllegalIndices (<[u:=ReferTo v]> dsu)
j < length (pathCompress (<[u:=ReferTo v]> dsu) n e v)
      exact (IH (<[u:=ReferTo v]> dsu) ltac:(rewrite insert_length; exact hV) h e ltac:(rewrite insert_length; exact (h1 _ _ he)) i _ hi).
    +
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j: nat
e: Tree
he: nth u dsu (Ancestor Unit) = Ancestor e
hi: nth i dsu (Ancestor Unit) = ReferTo j
j < length dsu pose proof h1 i j hi.
n, v: nat
IH: ∀ _0 : list Slot,
  v < length _0
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → noIllegalIndices
            (pathCompress _0 n _1 v)
dsu: list Slot
hV: v < length dsu
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
i, j: nat
e: Tree
he: nth u dsu (Ancestor Unit) = Ancestor e
hi: nth i dsu (Ancestor Unit) = ReferTo j
_0: j < length dsu
j < length dsu assumption.
Qed.

Lemma pathCompressPreservesWithoutCycles (dsu : list Slot) (h : withoutCyclesN dsu (length dsu)) (h1 : noIllegalIndices dsu) (n u : nat) (hU : u < length dsu) : withoutCyclesN (pathCompress dsu n u (ancestor dsu (length dsu) u)) (length dsu).
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
withoutCyclesN
  (pathCompress dsu n u (ancestor dsu (length dsu) u))
  (length dsu)
Proof.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
withoutCyclesN
  (pathCompress dsu n u (ancestor dsu (length dsu) u))
  (length dsu)
  induction n as [| n IH] in u, dsu, h, hU, h1 |- *.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
withoutCyclesN
  (pathCompress dsu 0 u (ancestor dsu (length dsu) u))
  (length dsu)
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
withoutCyclesN
  (pathCompress dsu (S n) u
     (ancestor dsu (length dsu) u)) (length dsu) {
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
withoutCyclesN
  (pathCompress dsu 0 u (ancestor dsu (length dsu) u))
  (length dsu) simpl.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
withoutCyclesN dsu (length dsu) assumption. }
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
withoutCyclesN
  (pathCompress dsu (S n) u
     (ancestor dsu (length dsu) u)) (length dsu)
  simpl.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
withoutCyclesN
  match nth u dsu (Ancestor Unit) with
  | ReferTo _0 =>
      pathCompress
        (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
           dsu) n _0 (ancestor dsu (length dsu) u)
  | Ancestor _ => dsu
  end (length dsu) remember (nth u dsu (Ancestor Unit)) as x eqn:hX.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: Slot
hX: x = nth u dsu (Ancestor Unit)
withoutCyclesN
  match x with
  | ReferTo _0 =>
      pathCompress
        (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
           dsu) n _0 (ancestor dsu (length dsu) u)
  | Ancestor _ => dsu
  end (length dsu) destruct x as [x | x]; [| easy].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) (length dsu) pose proof (fun t => IH (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu) t ltac:(intros v1 v2; destruct (decide (v1 = u)) as [h2 | h2]; [destruct (decide (length dsu <= u)); [rewrite list_insert_ge; [| lia]; intro h3; rewrite h2 in h3; rewrite <- hX in h3; injection h3; intro h4; subst x; subst u; exact (h1 v1 v2 ltac:(symmetry in hX; exact hX)) | pose proof list_lookup_insert dsu u (ReferTo (ancestor dsu (length dsu) u)) ltac:(lia) as step; pose proof nth_lookup_Some _ _ (Ancestor Unit) _ step as step2; subst u; rewrite step2; pose proof ancestorLtLength dsu h1 (length dsu) v1 ltac:(lia) as step3]; intro h3; injection h3; intro h4; rewrite insert_length | pose proof list_lookup_insert_ne dsu u v1 (ReferTo (ancestor dsu (length dsu) u)) ltac:(lia) as step; pose proof nth_lookup dsu v1 (Ancestor Unit) as step1; pose proof (nth_lookup (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu) v1 (Ancestor Unit)) as step2; rewrite step in step2; rewrite <- step1 in step2; rewrite step2; intro step3; rewrite insert_length; exact (h1 v1 v2 step3)]; lia) x ltac:(rewrite insert_length; exact (h1 u x ltac:(symmetry; assumption)))) as step.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu)
  (length
     (<[u:=ReferTo
             (ancestor dsu (length dsu) u)]>
        dsu))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor
          (<[u:=ReferTo
                  (ancestor dsu (length dsu) u)]>
             dsu)
          (length
             (<[u:=ReferTo
                     (ancestor dsu 
                        (length dsu) u)]> dsu))
          x))
    (length
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu))
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) (length dsu) rewrite !insert_length in step.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor
          (<[u:=ReferTo
                  (ancestor dsu (length dsu) u)]>
             dsu) (length dsu) x)) 
    (length dsu)
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
  assert (step1 : u < length dsu -> x < length dsu -> ancestor (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu) (length dsu) x = ancestor dsu (length dsu) x).
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor
          (<[u:=ReferTo
                  (ancestor dsu (length dsu) u)]>
             dsu) (length dsu) x)) 
    (length dsu)
u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor
          (<[u:=ReferTo
                  (ancestor dsu (length dsu) u)]>
             dsu) (length dsu) x)) 
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
  {
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor
          (<[u:=ReferTo
                  (ancestor dsu (length dsu) u)]>
             dsu) (length dsu) x)) 
    (length dsu)
u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x revert h h1 hX.
dsu: list Slot
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor
          (<[u:=ReferTo
                  (ancestor dsu (length dsu) u)]>
             dsu) (length dsu) x)) 
    (length dsu)
withoutCyclesN dsu (length dsu)
→ noIllegalIndices dsu
  → ReferTo x = nth u dsu (Ancestor Unit)
    → u < length dsu
      → x < length dsu
        → ancestor
            (<[u:=ReferTo
                    (ancestor dsu (length dsu) u)]>
               dsu) (length dsu) x =
          ancestor dsu (length dsu) x clear.
dsu: list Slot
u, x: nat
withoutCyclesN dsu (length dsu)
→ noIllegalIndices dsu
  → ReferTo x = nth u dsu (Ancestor Unit)
    → u < length dsu
      → x < length dsu
        → ancestor
            (<[u:=ReferTo
                    (ancestor dsu (length dsu) u)]>
               dsu) (length dsu) x =
          ancestor dsu (length dsu) x intros h1 h2 h3 h4 h5.
dsu: list Slot
u, x: nat
h1: withoutCyclesN dsu (length dsu)
h2: noIllegalIndices dsu
h3: ReferTo x = nth u dsu (Ancestor Unit)
h4: u < length dsu
h5: x < length dsu
ancestor
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
  (length dsu) x = ancestor dsu (length dsu) x rewrite <- ancestorInsert; try (assumption || reflexivity).
dsu: list Slot
u, x: nat
h1: withoutCyclesN dsu (length dsu)
h2: noIllegalIndices dsu
h3: ReferTo x = nth u dsu (Ancestor Unit)
h4: u < length dsu
h5: x < length dsu
match nth u dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end rewrite <- h3.
dsu: list Slot
u, x: nat
h1: withoutCyclesN dsu (length dsu)
h2: noIllegalIndices dsu
h3: ReferTo x = nth u dsu (Ancestor Unit)
h4: u < length dsu
h5: x < length dsu
true easy. }
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor
          (<[u:=ReferTo
                  (ancestor dsu (length dsu) u)]>
             dsu) (length dsu) x)) 
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu) rewrite step1 in step; try assumption; pose proof h1 u x ltac:(symmetry in hX; exact hX) as hX1; [| exact hX1].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
  remember (nth x dsu (Ancestor Unit)) as s eqn:hS.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Slot
hS: s = nth x dsu (Ancestor Unit)
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
  symmetry in hS.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Slot
hS: nth x dsu (Ancestor Unit) = s
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu) destruct s as [s | s].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
  -
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu) pose proof ancestorOfVertexInAncestorChain dsu (ancestorChain dsu (length dsu) u) u x h1 h hU hX1 ltac:(now rewrite hS) ltac:(split; [apply validChainAncestorChain; assumption | rewrite ancestorEqLastAncestorChain; pose proof h u hU; destruct (nth (ancestor dsu (length dsu) u)) as [| tree]; [easy | now exists tree]]) ltac:(destruct (length dsu); simpl; destruct (nth u dsu (Ancestor Unit)); simpl; try lia) 1 as step2.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu) rewrite <- step2 in step.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
nth 1 (ancestorChain dsu (length dsu) u) 0 = x
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
1 < length (ancestorChain dsu (length dsu) u)
    +
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu) apply step.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
  (length dsu) intros i hi.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
match
  nth
    (ancestor
       (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
          dsu)
       (length
          (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
             dsu)) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite insert_length, <- ancestorInsert; try assumption; [| now rewrite <- hX].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
      destruct (decide (ancestor dsu (length dsu) i = u)) as [hA | hA].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
hA: ancestor dsu (length dsu) i = u
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
hA: ancestor dsu (length dsu) i ≠ u
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
      *
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
hA: ancestor dsu (length dsu) i = u
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end pose proof h i hi as hj.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
hA: ancestor dsu (length dsu) i = u
hj: match
  nth (ancestor dsu (length dsu) i) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite hA, <- hX in hj.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
hA: ancestor dsu (length dsu) i = u
hj: false
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end exfalso.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
hA: ancestor dsu (length dsu) i = u
hj: false
False exact hj.
      *
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
hA: ancestor dsu (length dsu) i ≠ u
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite nth_lookup, list_lookup_insert_ne; [| lia].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
hA: ancestor dsu (length dsu) i ≠ u
match
  default (Ancestor Unit)
    (dsu !! ancestor dsu (length dsu) i)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite <- nth_lookup.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
hA: ancestor dsu (length dsu) i ≠ u
match
  nth (ancestor dsu (length dsu) i) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end apply h.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) u))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
i: nat
hi: i < length dsu
hA: ancestor dsu (length dsu) i ≠ u
i < length dsu assumption.
    +
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
nth 1 (ancestorChain dsu (length dsu) u) 0 = x assert (ht : 2 <= length dsu).
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
2 ≤ length dsu
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
nth 1 (ancestorChain dsu (length dsu) u) 0 = x
      {
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
2 ≤ length dsu destruct dsu as [| head tail].
h: withoutCyclesN [] (length [])
h1: noIllegalIndices []
n, u: nat
hU: u < length []
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor [] (length []) u)]>
     []) (length [])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [] (length []) u)]> [])
       n x (ancestor [] (length []) x))
    (length [])
step1: u < length []
→ x < length []
  → ancestor
      (<[u:=ReferTo
              (ancestor [] (length []) u)]> [])
      (length []) x =
    ancestor [] (length []) x
hX1: x < length []
s: nat
hS: nth x [] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [] (length []) u) 0 = x
→ 1 < length (ancestorChain [] (length []) u)
  → ancestor [] (length []) u =
    ancestor [] (length []) x
2 ≤ length []
head: Slot
tail: list Slot
h: withoutCyclesN (head :: tail)
  (length (head :: tail))
h1: noIllegalIndices (head :: tail)
n, u: nat
hU: u < length (head :: tail)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u (head :: tail) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: tail)
             (length (head :: tail)) u)]>
     (head :: tail)) (length (head :: tail))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: tail)
                  (length (head :: tail)) u)]>
          (head :: tail)) n x
       (ancestor (head :: tail)
          (length (head :: tail)) x))
    (length (head :: tail))
step1: u < length (head :: tail)
→ x < length (head :: tail)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: tail)
                 (length (head :: tail)) u)]>
         (head :: tail))
      (length (head :: tail)) x =
    ancestor (head :: tail)
      (length (head :: tail)) x
hX1: x < length (head :: tail)
s: nat
hS: nth x (head :: tail) (Ancestor Unit) = ReferTo s
step2: nth 1
  (ancestorChain (head :: tail)
     (length (head :: tail)) u) 0 = x
→ 1 <
  length
    (ancestorChain (head :: tail)
       (length (head :: tail)) u)
  → ancestor (head :: tail)
      (length (head :: tail)) u =
    ancestor (head :: tail)
      (length (head :: tail)) x
2 ≤ length (head :: tail) {
h: withoutCyclesN [] (length [])
h1: noIllegalIndices []
n, u: nat
hU: u < length []
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor [] (length []) u)]>
     []) (length [])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [] (length []) u)]> [])
       n x (ancestor [] (length []) x))
    (length [])
step1: u < length []
→ x < length []
  → ancestor
      (<[u:=ReferTo
              (ancestor [] (length []) u)]> [])
      (length []) x =
    ancestor [] (length []) x
hX1: x < length []
s: nat
hS: nth x [] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [] (length []) u) 0 = x
→ 1 < length (ancestorChain [] (length []) u)
  → ancestor [] (length []) u =
    ancestor [] (length []) x
2 ≤ length [] simpl in hU.
h: withoutCyclesN [] (length [])
h1: noIllegalIndices []
n, u: nat
hU: u < 0
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor [] (length []) u)]>
     []) (length [])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [] (length []) u)]> [])
       n x (ancestor [] (length []) x))
    (length [])
step1: u < length []
→ x < length []
  → ancestor
      (<[u:=ReferTo
              (ancestor [] (length []) u)]> [])
      (length []) x =
    ancestor [] (length []) x
hX1: x < length []
s: nat
hS: nth x [] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [] (length []) u) 0 = x
→ 1 < length (ancestorChain [] (length []) u)
  → ancestor [] (length []) u =
    ancestor [] (length []) x
2 ≤ length [] lia. }
head: Slot
tail: list Slot
h: withoutCyclesN (head :: tail)
  (length (head :: tail))
h1: noIllegalIndices (head :: tail)
n, u: nat
hU: u < length (head :: tail)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u (head :: tail) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: tail)
             (length (head :: tail)) u)]>
     (head :: tail)) (length (head :: tail))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: tail)
                  (length (head :: tail)) u)]>
          (head :: tail)) n x
       (ancestor (head :: tail)
          (length (head :: tail)) x))
    (length (head :: tail))
step1: u < length (head :: tail)
→ x < length (head :: tail)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: tail)
                 (length (head :: tail)) u)]>
         (head :: tail))
      (length (head :: tail)) x =
    ancestor (head :: tail)
      (length (head :: tail)) x
hX1: x < length (head :: tail)
s: nat
hS: nth x (head :: tail) (Ancestor Unit) = ReferTo s
step2: nth 1
  (ancestorChain (head :: tail)
     (length (head :: tail)) u) 0 = x
→ 1 <
  length
    (ancestorChain (head :: tail)
       (length (head :: tail)) u)
  → ancestor (head :: tail)
      (length (head :: tail)) u =
    ancestor (head :: tail)
      (length (head :: tail)) x
2 ≤ length (head :: tail)
        destruct tail.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < length [head]
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) u)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) u)
  → ancestor [head] (length [head]) u =
    ancestor [head] (length [head]) x
2 ≤ length [head]
head, _0: Slot
_1: list Slot
h: withoutCyclesN (head :: _0 :: _1)
  (length (head :: _0 :: _1))
h1: noIllegalIndices (head :: _0 :: _1)
n, u: nat
hU: u < length (head :: _0 :: _1)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x =
nth u (head :: _0 :: _1) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: _0 :: _1)
             (length (head :: _0 :: _1)) u)]>
     (head :: _0 :: _1))
  (length (head :: _0 :: _1))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: _0 :: _1)
                  (length (head :: _0 :: _1)) u)]>
          (head :: _0 :: _1)) n x
       (ancestor (head :: _0 :: _1)
          (length (head :: _0 :: _1)) x))
    (length (head :: _0 :: _1))
step1: u < length (head :: _0 :: _1)
→ x < length (head :: _0 :: _1)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: _0 :: _1)
                 (length (head :: _0 :: _1)) u)]>
         (head :: _0 :: _1))
      (length (head :: _0 :: _1)) x =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
hX1: x < length (head :: _0 :: _1)
s: nat
hS: nth x (head :: _0 :: _1) (Ancestor Unit) =
ReferTo s
step2: nth 1
  (ancestorChain (head :: _0 :: _1)
     (length (head :: _0 :: _1)) u) 0 = x
→ 1 <
  length
    (ancestorChain (head :: _0 :: _1)
       (length (head :: _0 :: _1)) u)
  → ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) u =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
2 ≤ length (head :: _0 :: _1) {
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < length [head]
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) u)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) u)
  → ancestor [head] (length [head]) u =
    ancestor [head] (length [head]) x
2 ≤ length [head] simpl in hU.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) u)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) u)
  → ancestor [head] (length [head]) u =
    ancestor [head] (length [head]) x
2 ≤ length [head] pose proof (ltac:(lia) : u = 0).
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) u)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) u)
  → ancestor [head] (length [head]) u =
    ancestor [head] (length [head]) x
_0: u = 0
2 ≤ length [head] subst u.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
step1: 0 < length [head]
→ x < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
hX: ReferTo x = nth 0 [head] (Ancestor Unit)
hX1: x < length [head]
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) x
2 ≤ length [head] simpl in hX, hX1.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
step1: 0 < length [head]
→ x < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
hX: ReferTo x = head
hX1: x < 1
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) x
2 ≤ length [head] pose proof (ltac:(lia) : x = 0).
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
step1: 0 < length [head]
→ x < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
hX: ReferTo x = head
hX1: x < 1
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) x
_0: x = 0
2 ≤ length [head] subst x.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
hX: ReferTo 0 = head
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n 0 (ancestor [head] (length [head]) 0))
    (length [head])
step1: 0 < length [head]
→ 0 < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) 0 =
    ancestor [head] (length [head]) 0
s: nat
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = 0
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) 0
hS: nth 0 [head] (Ancestor Unit) = ReferTo s
2 ≤ length [head] pose proof h 0 ltac:(simpl; lia) as step3.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
hX: ReferTo 0 = head
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n 0 (ancestor [head] (length [head]) 0))
    (length [head])
step1: 0 < length [head]
→ 0 < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) 0 =
    ancestor [head] (length [head]) 0
s: nat
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = 0
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) 0
hS: nth 0 [head] (Ancestor Unit) = ReferTo s
step3: match
  nth (ancestor [head] (length [head]) 0)
    [head] (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
2 ≤ length [head] simpl in step3, hS.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
hX: ReferTo 0 = head
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n 0 (ancestor [head] (length [head]) 0))
    (length [head])
step1: 0 < length [head]
→ 0 < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) 0 =
    ancestor [head] (length [head]) 0
s: nat
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = 0
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) 0
hS: head = ReferTo s
step3: match
  match
    match head with
    | ReferTo _0 => _0
    | Ancestor _ => 0
    end
  with
  | 0 => head
  | _ => Ancestor Unit
  end
with
| ReferTo _ => false
| Ancestor _ => true
end
2 ≤ length [head] subst head.
h1: noIllegalIndices [ReferTo 0]
h: withoutCyclesN [ReferTo 0] (length [ReferTo 0])
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
step1: 0 < length [ReferTo 0]
→ 0 < length [ReferTo 0]
  → ancestor
      (<[0:=ReferTo
              (ancestor [
                 ReferTo 0]
                 (length [ReferTo 0]) 0)]>
         [ReferTo 0]) (length [ReferTo 0]) 0 =
    ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [ReferTo 0]
             (length [ReferTo 0]) 0)]>
     [ReferTo 0]) (length [ReferTo 0])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [
                  ReferTo 0]
                  (length [ReferTo 0]) 0)]>
          [ReferTo 0]) n 0
       (ancestor [ReferTo 0]
          (length [ReferTo 0]) 0))
    (length [ReferTo 0])
s: nat
step3: false
hS: ReferTo 0 = ReferTo s
step2: nth 1
  (ancestorChain [ReferTo 0]
     (length [ReferTo 0]) 0) 0 = 0
→ 1 <
  length
    (ancestorChain [ReferTo 0]
       (length [ReferTo 0]) 0)
  → ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0 =
    ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0
2 ≤ length [ReferTo 0] exfalso.
h1: noIllegalIndices [ReferTo 0]
h: withoutCyclesN [ReferTo 0] (length [ReferTo 0])
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
step1: 0 < length [ReferTo 0]
→ 0 < length [ReferTo 0]
  → ancestor
      (<[0:=ReferTo
              (ancestor [
                 ReferTo 0]
                 (length [ReferTo 0]) 0)]>
         [ReferTo 0]) (length [ReferTo 0]) 0 =
    ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [ReferTo 0]
             (length [ReferTo 0]) 0)]>
     [ReferTo 0]) (length [ReferTo 0])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [
                  ReferTo 0]
                  (length [ReferTo 0]) 0)]>
          [ReferTo 0]) n 0
       (ancestor [ReferTo 0]
          (length [ReferTo 0]) 0))
    (length [ReferTo 0])
s: nat
step3: false
hS: ReferTo 0 = ReferTo s
step2: nth 1
  (ancestorChain [ReferTo 0]
     (length [ReferTo 0]) 0) 0 = 0
→ 1 <
  length
    (ancestorChain [ReferTo 0]
       (length [ReferTo 0]) 0)
  → ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0 =
    ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0
False exact step3. }
head, _0: Slot
_1: list Slot
h: withoutCyclesN (head :: _0 :: _1)
  (length (head :: _0 :: _1))
h1: noIllegalIndices (head :: _0 :: _1)
n, u: nat
hU: u < length (head :: _0 :: _1)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x =
nth u (head :: _0 :: _1) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: _0 :: _1)
             (length (head :: _0 :: _1)) u)]>
     (head :: _0 :: _1))
  (length (head :: _0 :: _1))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: _0 :: _1)
                  (length (head :: _0 :: _1)) u)]>
          (head :: _0 :: _1)) n x
       (ancestor (head :: _0 :: _1)
          (length (head :: _0 :: _1)) x))
    (length (head :: _0 :: _1))
step1: u < length (head :: _0 :: _1)
→ x < length (head :: _0 :: _1)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: _0 :: _1)
                 (length (head :: _0 :: _1)) u)]>
         (head :: _0 :: _1))
      (length (head :: _0 :: _1)) x =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
hX1: x < length (head :: _0 :: _1)
s: nat
hS: nth x (head :: _0 :: _1) (Ancestor Unit) =
ReferTo s
step2: nth 1
  (ancestorChain (head :: _0 :: _1)
     (length (head :: _0 :: _1)) u) 0 = x
→ 1 <
  length
    (ancestorChain (head :: _0 :: _1)
       (length (head :: _0 :: _1)) u)
  → ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) u =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
2 ≤ length (head :: _0 :: _1) simpl.
head, _0: Slot
_1: list Slot
h: withoutCyclesN (head :: _0 :: _1)
  (length (head :: _0 :: _1))
h1: noIllegalIndices (head :: _0 :: _1)
n, u: nat
hU: u < length (head :: _0 :: _1)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x =
nth u (head :: _0 :: _1) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: _0 :: _1)
             (length (head :: _0 :: _1)) u)]>
     (head :: _0 :: _1))
  (length (head :: _0 :: _1))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: _0 :: _1)
                  (length (head :: _0 :: _1)) u)]>
          (head :: _0 :: _1)) n x
       (ancestor (head :: _0 :: _1)
          (length (head :: _0 :: _1)) x))
    (length (head :: _0 :: _1))
step1: u < length (head :: _0 :: _1)
→ x < length (head :: _0 :: _1)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: _0 :: _1)
                 (length (head :: _0 :: _1)) u)]>
         (head :: _0 :: _1))
      (length (head :: _0 :: _1)) x =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
hX1: x < length (head :: _0 :: _1)
s: nat
hS: nth x (head :: _0 :: _1) (Ancestor Unit) =
ReferTo s
step2: nth 1
  (ancestorChain (head :: _0 :: _1)
     (length (head :: _0 :: _1)) u) 0 = x
→ 1 <
  length
    (ancestorChain (head :: _0 :: _1)
       (length (head :: _0 :: _1)) u)
  → ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) u =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
2 ≤ S (S (length _1)) lia. }
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
nth 1 (ancestorChain dsu (length dsu) u) 0 = x
      destruct (ltac:(exists (length dsu - 2); lia) : exists n, length dsu = S (S n)) as [nn hn].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
nth 1 (ancestorChain dsu (length dsu) u) 0 = x rewrite hn.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
nth 1 (ancestorChain dsu (S (S nn)) u) 0 = x simpl.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
nth 1
  match nth u dsu (Ancestor Unit) with
  | ReferTo _0 =>
      u
      :: match nth _0 dsu (Ancestor Unit) with
         | ReferTo _1 => _0 :: ancestorChain dsu nn _1
         | Ancestor _ => [_0]
         end
  | Ancestor _ => [u]
  end 0 = x rewrite <- hX, hS.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
nth 1 (u :: x :: ancestorChain dsu nn s) 0 = x easy.
    +
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
1 < length (ancestorChain dsu (length dsu) u) assert (ht : 2 <= length dsu).
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
2 ≤ length dsu
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
1 < length (ancestorChain dsu (length dsu) u)
      {
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
2 ≤ length dsu destruct dsu as [| head tail].
h: withoutCyclesN [] (length [])
h1: noIllegalIndices []
n, u: nat
hU: u < length []
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor [] (length []) u)]>
     []) (length [])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [] (length []) u)]> [])
       n x (ancestor [] (length []) x))
    (length [])
step1: u < length []
→ x < length []
  → ancestor
      (<[u:=ReferTo
              (ancestor [] (length []) u)]> [])
      (length []) x =
    ancestor [] (length []) x
hX1: x < length []
s: nat
hS: nth x [] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [] (length []) u) 0 = x
→ 1 < length (ancestorChain [] (length []) u)
  → ancestor [] (length []) u =
    ancestor [] (length []) x
2 ≤ length []
head: Slot
tail: list Slot
h: withoutCyclesN (head :: tail)
  (length (head :: tail))
h1: noIllegalIndices (head :: tail)
n, u: nat
hU: u < length (head :: tail)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u (head :: tail) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: tail)
             (length (head :: tail)) u)]>
     (head :: tail)) (length (head :: tail))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: tail)
                  (length (head :: tail)) u)]>
          (head :: tail)) n x
       (ancestor (head :: tail)
          (length (head :: tail)) x))
    (length (head :: tail))
step1: u < length (head :: tail)
→ x < length (head :: tail)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: tail)
                 (length (head :: tail)) u)]>
         (head :: tail))
      (length (head :: tail)) x =
    ancestor (head :: tail)
      (length (head :: tail)) x
hX1: x < length (head :: tail)
s: nat
hS: nth x (head :: tail) (Ancestor Unit) = ReferTo s
step2: nth 1
  (ancestorChain (head :: tail)
     (length (head :: tail)) u) 0 = x
→ 1 <
  length
    (ancestorChain (head :: tail)
       (length (head :: tail)) u)
  → ancestor (head :: tail)
      (length (head :: tail)) u =
    ancestor (head :: tail)
      (length (head :: tail)) x
2 ≤ length (head :: tail) {
h: withoutCyclesN [] (length [])
h1: noIllegalIndices []
n, u: nat
hU: u < length []
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor [] (length []) u)]>
     []) (length [])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [] (length []) u)]> [])
       n x (ancestor [] (length []) x))
    (length [])
step1: u < length []
→ x < length []
  → ancestor
      (<[u:=ReferTo
              (ancestor [] (length []) u)]> [])
      (length []) x =
    ancestor [] (length []) x
hX1: x < length []
s: nat
hS: nth x [] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [] (length []) u) 0 = x
→ 1 < length (ancestorChain [] (length []) u)
  → ancestor [] (length []) u =
    ancestor [] (length []) x
2 ≤ length [] simpl in hU.
h: withoutCyclesN [] (length [])
h1: noIllegalIndices []
n, u: nat
hU: u < 0
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor [] (length []) u)]>
     []) (length [])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [] (length []) u)]> [])
       n x (ancestor [] (length []) x))
    (length [])
step1: u < length []
→ x < length []
  → ancestor
      (<[u:=ReferTo
              (ancestor [] (length []) u)]> [])
      (length []) x =
    ancestor [] (length []) x
hX1: x < length []
s: nat
hS: nth x [] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [] (length []) u) 0 = x
→ 1 < length (ancestorChain [] (length []) u)
  → ancestor [] (length []) u =
    ancestor [] (length []) x
2 ≤ length [] lia. }
head: Slot
tail: list Slot
h: withoutCyclesN (head :: tail)
  (length (head :: tail))
h1: noIllegalIndices (head :: tail)
n, u: nat
hU: u < length (head :: tail)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u (head :: tail) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: tail)
             (length (head :: tail)) u)]>
     (head :: tail)) (length (head :: tail))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: tail)
                  (length (head :: tail)) u)]>
          (head :: tail)) n x
       (ancestor (head :: tail)
          (length (head :: tail)) x))
    (length (head :: tail))
step1: u < length (head :: tail)
→ x < length (head :: tail)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: tail)
                 (length (head :: tail)) u)]>
         (head :: tail))
      (length (head :: tail)) x =
    ancestor (head :: tail)
      (length (head :: tail)) x
hX1: x < length (head :: tail)
s: nat
hS: nth x (head :: tail) (Ancestor Unit) = ReferTo s
step2: nth 1
  (ancestorChain (head :: tail)
     (length (head :: tail)) u) 0 = x
→ 1 <
  length
    (ancestorChain (head :: tail)
       (length (head :: tail)) u)
  → ancestor (head :: tail)
      (length (head :: tail)) u =
    ancestor (head :: tail)
      (length (head :: tail)) x
2 ≤ length (head :: tail)
        destruct tail.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < length [head]
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) u)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) u)
  → ancestor [head] (length [head]) u =
    ancestor [head] (length [head]) x
2 ≤ length [head]
head, _0: Slot
_1: list Slot
h: withoutCyclesN (head :: _0 :: _1)
  (length (head :: _0 :: _1))
h1: noIllegalIndices (head :: _0 :: _1)
n, u: nat
hU: u < length (head :: _0 :: _1)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x =
nth u (head :: _0 :: _1) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: _0 :: _1)
             (length (head :: _0 :: _1)) u)]>
     (head :: _0 :: _1))
  (length (head :: _0 :: _1))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: _0 :: _1)
                  (length (head :: _0 :: _1)) u)]>
          (head :: _0 :: _1)) n x
       (ancestor (head :: _0 :: _1)
          (length (head :: _0 :: _1)) x))
    (length (head :: _0 :: _1))
step1: u < length (head :: _0 :: _1)
→ x < length (head :: _0 :: _1)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: _0 :: _1)
                 (length (head :: _0 :: _1)) u)]>
         (head :: _0 :: _1))
      (length (head :: _0 :: _1)) x =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
hX1: x < length (head :: _0 :: _1)
s: nat
hS: nth x (head :: _0 :: _1) (Ancestor Unit) =
ReferTo s
step2: nth 1
  (ancestorChain (head :: _0 :: _1)
     (length (head :: _0 :: _1)) u) 0 = x
→ 1 <
  length
    (ancestorChain (head :: _0 :: _1)
       (length (head :: _0 :: _1)) u)
  → ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) u =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
2 ≤ length (head :: _0 :: _1) {
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < length [head]
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) u)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) u)
  → ancestor [head] (length [head]) u =
    ancestor [head] (length [head]) x
2 ≤ length [head] simpl in hU.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) u)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) u)
  → ancestor [head] (length [head]) u =
    ancestor [head] (length [head]) x
2 ≤ length [head] pose proof (ltac:(lia) : u = 0).
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) u)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) u)
  → ancestor [head] (length [head]) u =
    ancestor [head] (length [head]) x
_0: u = 0
2 ≤ length [head] subst u.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
step1: 0 < length [head]
→ x < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
hX: ReferTo x = nth 0 [head] (Ancestor Unit)
hX1: x < length [head]
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) x
2 ≤ length [head] simpl in hX, hX1.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
step1: 0 < length [head]
→ x < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
hX: ReferTo x = head
hX1: x < 1
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) x
2 ≤ length [head] pose proof (ltac:(lia) : x = 0).
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
step1: 0 < length [head]
→ x < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
hX: ReferTo x = head
hX1: x < 1
s: nat
hS: nth x [head] (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = x
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) x
_0: x = 0
2 ≤ length [head] subst x.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
hX: ReferTo 0 = head
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n 0 (ancestor [head] (length [head]) 0))
    (length [head])
step1: 0 < length [head]
→ 0 < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) 0 =
    ancestor [head] (length [head]) 0
s: nat
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = 0
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) 0
hS: nth 0 [head] (Ancestor Unit) = ReferTo s
2 ≤ length [head] pose proof h 0 ltac:(simpl; lia) as step3.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
hX: ReferTo 0 = head
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n 0 (ancestor [head] (length [head]) 0))
    (length [head])
step1: 0 < length [head]
→ 0 < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) 0 =
    ancestor [head] (length [head]) 0
s: nat
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = 0
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) 0
hS: nth 0 [head] (Ancestor Unit) = ReferTo s
step3: match
  nth (ancestor [head] (length [head]) 0)
    [head] (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
2 ≤ length [head] simpl in step3, hS.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
hX: ReferTo 0 = head
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n 0 (ancestor [head] (length [head]) 0))
    (length [head])
step1: 0 < length [head]
→ 0 < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) 0 =
    ancestor [head] (length [head]) 0
s: nat
step2: nth 1 (ancestorChain [head] (length [head]) 0)
  0 = 0
→ 1 <
  length
    (ancestorChain [head] (length [head]) 0)
  → ancestor [head] (length [head]) 0 =
    ancestor [head] (length [head]) 0
hS: head = ReferTo s
step3: match
  match
    match head with
    | ReferTo _0 => _0
    | Ancestor _ => 0
    end
  with
  | 0 => head
  | _ => Ancestor Unit
  end
with
| ReferTo _ => false
| Ancestor _ => true
end
2 ≤ length [head] subst head.
h1: noIllegalIndices [ReferTo 0]
h: withoutCyclesN [ReferTo 0] (length [ReferTo 0])
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
step1: 0 < length [ReferTo 0]
→ 0 < length [ReferTo 0]
  → ancestor
      (<[0:=ReferTo
              (ancestor [
                 ReferTo 0]
                 (length [ReferTo 0]) 0)]>
         [ReferTo 0]) (length [ReferTo 0]) 0 =
    ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [ReferTo 0]
             (length [ReferTo 0]) 0)]>
     [ReferTo 0]) (length [ReferTo 0])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [
                  ReferTo 0]
                  (length [ReferTo 0]) 0)]>
          [ReferTo 0]) n 0
       (ancestor [ReferTo 0]
          (length [ReferTo 0]) 0))
    (length [ReferTo 0])
s: nat
step3: false
hS: ReferTo 0 = ReferTo s
step2: nth 1
  (ancestorChain [ReferTo 0]
     (length [ReferTo 0]) 0) 0 = 0
→ 1 <
  length
    (ancestorChain [ReferTo 0]
       (length [ReferTo 0]) 0)
  → ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0 =
    ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0
2 ≤ length [ReferTo 0] exfalso.
h1: noIllegalIndices [ReferTo 0]
h: withoutCyclesN [ReferTo 0] (length [ReferTo 0])
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
step1: 0 < length [ReferTo 0]
→ 0 < length [ReferTo 0]
  → ancestor
      (<[0:=ReferTo
              (ancestor [
                 ReferTo 0]
                 (length [ReferTo 0]) 0)]>
         [ReferTo 0]) (length [ReferTo 0]) 0 =
    ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [ReferTo 0]
             (length [ReferTo 0]) 0)]>
     [ReferTo 0]) (length [ReferTo 0])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [
                  ReferTo 0]
                  (length [ReferTo 0]) 0)]>
          [ReferTo 0]) n 0
       (ancestor [ReferTo 0]
          (length [ReferTo 0]) 0))
    (length [ReferTo 0])
s: nat
step3: false
hS: ReferTo 0 = ReferTo s
step2: nth 1
  (ancestorChain [ReferTo 0]
     (length [ReferTo 0]) 0) 0 = 0
→ 1 <
  length
    (ancestorChain [ReferTo 0]
       (length [ReferTo 0]) 0)
  → ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0 =
    ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0
False exact step3. }
head, _0: Slot
_1: list Slot
h: withoutCyclesN (head :: _0 :: _1)
  (length (head :: _0 :: _1))
h1: noIllegalIndices (head :: _0 :: _1)
n, u: nat
hU: u < length (head :: _0 :: _1)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x =
nth u (head :: _0 :: _1) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: _0 :: _1)
             (length (head :: _0 :: _1)) u)]>
     (head :: _0 :: _1))
  (length (head :: _0 :: _1))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: _0 :: _1)
                  (length (head :: _0 :: _1)) u)]>
          (head :: _0 :: _1)) n x
       (ancestor (head :: _0 :: _1)
          (length (head :: _0 :: _1)) x))
    (length (head :: _0 :: _1))
step1: u < length (head :: _0 :: _1)
→ x < length (head :: _0 :: _1)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: _0 :: _1)
                 (length (head :: _0 :: _1)) u)]>
         (head :: _0 :: _1))
      (length (head :: _0 :: _1)) x =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
hX1: x < length (head :: _0 :: _1)
s: nat
hS: nth x (head :: _0 :: _1) (Ancestor Unit) =
ReferTo s
step2: nth 1
  (ancestorChain (head :: _0 :: _1)
     (length (head :: _0 :: _1)) u) 0 = x
→ 1 <
  length
    (ancestorChain (head :: _0 :: _1)
       (length (head :: _0 :: _1)) u)
  → ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) u =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
2 ≤ length (head :: _0 :: _1) simpl.
head, _0: Slot
_1: list Slot
h: withoutCyclesN (head :: _0 :: _1)
  (length (head :: _0 :: _1))
h1: noIllegalIndices (head :: _0 :: _1)
n, u: nat
hU: u < length (head :: _0 :: _1)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x =
nth u (head :: _0 :: _1) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: _0 :: _1)
             (length (head :: _0 :: _1)) u)]>
     (head :: _0 :: _1))
  (length (head :: _0 :: _1))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: _0 :: _1)
                  (length (head :: _0 :: _1)) u)]>
          (head :: _0 :: _1)) n x
       (ancestor (head :: _0 :: _1)
          (length (head :: _0 :: _1)) x))
    (length (head :: _0 :: _1))
step1: u < length (head :: _0 :: _1)
→ x < length (head :: _0 :: _1)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: _0 :: _1)
                 (length (head :: _0 :: _1)) u)]>
         (head :: _0 :: _1))
      (length (head :: _0 :: _1)) x =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
hX1: x < length (head :: _0 :: _1)
s: nat
hS: nth x (head :: _0 :: _1) (Ancestor Unit) =
ReferTo s
step2: nth 1
  (ancestorChain (head :: _0 :: _1)
     (length (head :: _0 :: _1)) u) 0 = x
→ 1 <
  length
    (ancestorChain (head :: _0 :: _1)
       (length (head :: _0 :: _1)) u)
  → ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) u =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
2 ≤ S (S (length _1)) lia. }
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
1 < length (ancestorChain dsu (length dsu) u)
      destruct (ltac:(exists (length dsu - 2); lia) : exists n, length dsu = S (S n)) as [nn hn].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
1 < length (ancestorChain dsu (length dsu) u) rewrite hn.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
1 < length (ancestorChain dsu (S (S nn)) u) simpl.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
1 <
length
  match nth u dsu (Ancestor Unit) with
  | ReferTo _0 =>
      u
      :: match nth _0 dsu (Ancestor Unit) with
         | ReferTo _1 => _0 :: ancestorChain dsu nn _1
         | Ancestor _ => [_0]
         end
  | Ancestor _ => [u]
  end rewrite <- hX, hS.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
1 < length (u :: x :: ancestorChain dsu nn s) simpl.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: nat
hS: nth x dsu (Ancestor Unit) = ReferTo s
step2: nth 1 (ancestorChain dsu (length dsu) u) 0 = x
→ 1 <
  length (ancestorChain dsu (length dsu) u)
  → ancestor dsu (length dsu) u =
    ancestor dsu (length dsu) x
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
1 < S (S (length (ancestorChain dsu nn s))) lia.
  -
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu) assert (ht : 2 <= length dsu).
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
2 ≤ length dsu
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
    {
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
2 ≤ length dsu destruct dsu as [| head tail].
h: withoutCyclesN [] (length [])
h1: noIllegalIndices []
n, u: nat
hU: u < length []
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor [] (length []) u)]>
     []) (length [])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [] (length []) u)]> [])
       n x (ancestor [] (length []) x))
    (length [])
step1: u < length []
→ x < length []
  → ancestor
      (<[u:=ReferTo
              (ancestor [] (length []) u)]> [])
      (length []) x =
    ancestor [] (length []) x
hX1: x < length []
s: Tree
hS: nth x [] (Ancestor Unit) = Ancestor s
2 ≤ length []
head: Slot
tail: list Slot
h: withoutCyclesN (head :: tail)
  (length (head :: tail))
h1: noIllegalIndices (head :: tail)
n, u: nat
hU: u < length (head :: tail)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u (head :: tail) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: tail)
             (length (head :: tail)) u)]>
     (head :: tail)) (length (head :: tail))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: tail)
                  (length (head :: tail)) u)]>
          (head :: tail)) n x
       (ancestor (head :: tail)
          (length (head :: tail)) x))
    (length (head :: tail))
step1: u < length (head :: tail)
→ x < length (head :: tail)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: tail)
                 (length (head :: tail)) u)]>
         (head :: tail))
      (length (head :: tail)) x =
    ancestor (head :: tail)
      (length (head :: tail)) x
hX1: x < length (head :: tail)
s: Tree
hS: nth x (head :: tail) (Ancestor Unit) = Ancestor s
2 ≤ length (head :: tail) {
h: withoutCyclesN [] (length [])
h1: noIllegalIndices []
n, u: nat
hU: u < length []
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor [] (length []) u)]>
     []) (length [])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [] (length []) u)]> [])
       n x (ancestor [] (length []) x))
    (length [])
step1: u < length []
→ x < length []
  → ancestor
      (<[u:=ReferTo
              (ancestor [] (length []) u)]> [])
      (length []) x =
    ancestor [] (length []) x
hX1: x < length []
s: Tree
hS: nth x [] (Ancestor Unit) = Ancestor s
2 ≤ length [] simpl in hU.
h: withoutCyclesN [] (length [])
h1: noIllegalIndices []
n, u: nat
hU: u < 0
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor [] (length []) u)]>
     []) (length [])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [] (length []) u)]> [])
       n x (ancestor [] (length []) x))
    (length [])
step1: u < length []
→ x < length []
  → ancestor
      (<[u:=ReferTo
              (ancestor [] (length []) u)]> [])
      (length []) x =
    ancestor [] (length []) x
hX1: x < length []
s: Tree
hS: nth x [] (Ancestor Unit) = Ancestor s
2 ≤ length [] lia. }
head: Slot
tail: list Slot
h: withoutCyclesN (head :: tail)
  (length (head :: tail))
h1: noIllegalIndices (head :: tail)
n, u: nat
hU: u < length (head :: tail)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u (head :: tail) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: tail)
             (length (head :: tail)) u)]>
     (head :: tail)) (length (head :: tail))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: tail)
                  (length (head :: tail)) u)]>
          (head :: tail)) n x
       (ancestor (head :: tail)
          (length (head :: tail)) x))
    (length (head :: tail))
step1: u < length (head :: tail)
→ x < length (head :: tail)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: tail)
                 (length (head :: tail)) u)]>
         (head :: tail))
      (length (head :: tail)) x =
    ancestor (head :: tail)
      (length (head :: tail)) x
hX1: x < length (head :: tail)
s: Tree
hS: nth x (head :: tail) (Ancestor Unit) = Ancestor s
2 ≤ length (head :: tail)
      destruct tail.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < length [head]
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: Tree
hS: nth x [head] (Ancestor Unit) = Ancestor s
2 ≤ length [head]
head, _0: Slot
_1: list Slot
h: withoutCyclesN (head :: _0 :: _1)
  (length (head :: _0 :: _1))
h1: noIllegalIndices (head :: _0 :: _1)
n, u: nat
hU: u < length (head :: _0 :: _1)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x =
nth u (head :: _0 :: _1) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: _0 :: _1)
             (length (head :: _0 :: _1)) u)]>
     (head :: _0 :: _1))
  (length (head :: _0 :: _1))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: _0 :: _1)
                  (length (head :: _0 :: _1)) u)]>
          (head :: _0 :: _1)) n x
       (ancestor (head :: _0 :: _1)
          (length (head :: _0 :: _1)) x))
    (length (head :: _0 :: _1))
step1: u < length (head :: _0 :: _1)
→ x < length (head :: _0 :: _1)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: _0 :: _1)
                 (length (head :: _0 :: _1)) u)]>
         (head :: _0 :: _1))
      (length (head :: _0 :: _1)) x =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
hX1: x < length (head :: _0 :: _1)
s: Tree
hS: nth x (head :: _0 :: _1) (Ancestor Unit) =
Ancestor s
2 ≤ length (head :: _0 :: _1) {
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < length [head]
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: Tree
hS: nth x [head] (Ancestor Unit) = Ancestor s
2 ≤ length [head] simpl in hU.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: Tree
hS: nth x [head] (Ancestor Unit) = Ancestor s
2 ≤ length [head] pose proof (ltac:(lia) : u = 0).
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n, u: nat
hU: u < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u [head] (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor [head] (length [head]) u)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor [head] 
                  (length [head]) u)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
step1: u < length [head]
→ x < length [head]
  → ancestor
      (<[u:=ReferTo
              (ancestor [head] 
                 (length [head]) u)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
hX1: x < length [head]
s: Tree
hS: nth x [head] (Ancestor Unit) = Ancestor s
_0: u = 0
2 ≤ length [head] subst u.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
step1: 0 < length [head]
→ x < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
hX: ReferTo x = nth 0 [head] (Ancestor Unit)
hX1: x < length [head]
s: Tree
hS: nth x [head] (Ancestor Unit) = Ancestor s
2 ≤ length [head] simpl in hX, hX1.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
step1: 0 < length [head]
→ x < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
hX: ReferTo x = head
hX1: x < 1
s: Tree
hS: nth x [head] (Ancestor Unit) = Ancestor s
2 ≤ length [head] pose proof (ltac:(lia) : x = 0).
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
step1: 0 < length [head]
→ x < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) x =
    ancestor [head] (length [head]) x
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n x (ancestor [head] (length [head]) x))
    (length [head])
hX: ReferTo x = head
hX1: x < 1
s: Tree
hS: nth x [head] (Ancestor Unit) = Ancestor s
_0: x = 0
2 ≤ length [head] subst x.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
hX: ReferTo 0 = head
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n 0 (ancestor [head] (length [head]) 0))
    (length [head])
step1: 0 < length [head]
→ 0 < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) 0 =
    ancestor [head] (length [head]) 0
s: Tree
hS: nth 0 [head] (Ancestor Unit) = Ancestor s
2 ≤ length [head] pose proof h 0 ltac:(simpl; lia) as step3.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
hX: ReferTo 0 = head
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n 0 (ancestor [head] (length [head]) 0))
    (length [head])
step1: 0 < length [head]
→ 0 < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) 0 =
    ancestor [head] (length [head]) 0
s: Tree
hS: nth 0 [head] (Ancestor Unit) = Ancestor s
step3: match
  nth (ancestor [head] (length [head]) 0)
    [head] (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
2 ≤ length [head] simpl in step3, hS.
head: Slot
h: withoutCyclesN [head] (length [head])
h1: noIllegalIndices [head]
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
hX: ReferTo 0 = head
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [head] (length [head]) 0)]>
     [head]) (length [head])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [head] 
                  (length [head]) 0)]> [head])
       n 0 (ancestor [head] (length [head]) 0))
    (length [head])
step1: 0 < length [head]
→ 0 < length [head]
  → ancestor
      (<[0:=ReferTo
              (ancestor [head] 
                 (length [head]) 0)]> [head])
      (length [head]) 0 =
    ancestor [head] (length [head]) 0
s: Tree
hS: head = Ancestor s
step3: match
  match
    match head with
    | ReferTo _0 => _0
    | Ancestor _ => 0
    end
  with
  | 0 => head
  | _ => Ancestor Unit
  end
with
| ReferTo _ => false
| Ancestor _ => true
end
2 ≤ length [head] subst head.
h1: noIllegalIndices [ReferTo 0]
h: withoutCyclesN [ReferTo 0] (length [ReferTo 0])
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
step1: 0 < length [ReferTo 0]
→ 0 < length [ReferTo 0]
  → ancestor
      (<[0:=ReferTo
              (ancestor [
                 ReferTo 0]
                 (length [ReferTo 0]) 0)]>
         [ReferTo 0]) (length [ReferTo 0]) 0 =
    ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [ReferTo 0]
             (length [ReferTo 0]) 0)]>
     [ReferTo 0]) (length [ReferTo 0])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [
                  ReferTo 0]
                  (length [ReferTo 0]) 0)]>
          [ReferTo 0]) n 0
       (ancestor [ReferTo 0]
          (length [ReferTo 0]) 0))
    (length [ReferTo 0])
s: Tree
step3: false
hS: ReferTo 0 = Ancestor s
2 ≤ length [ReferTo 0] exfalso.
h1: noIllegalIndices [ReferTo 0]
h: withoutCyclesN [ReferTo 0] (length [ReferTo 0])
n: nat
hU: 0 < 1
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
hX1: 0 < 1
step1: 0 < length [ReferTo 0]
→ 0 < length [ReferTo 0]
  → ancestor
      (<[0:=ReferTo
              (ancestor [
                 ReferTo 0]
                 (length [ReferTo 0]) 0)]>
         [ReferTo 0]) (length [ReferTo 0]) 0 =
    ancestor [ReferTo 0] 
      (length [ReferTo 0]) 0
step: withoutCyclesN
  (<[0:=ReferTo
          (ancestor [ReferTo 0]
             (length [ReferTo 0]) 0)]>
     [ReferTo 0]) (length [ReferTo 0])
→ withoutCyclesN
    (pathCompress
       (<[0:=ReferTo
               (ancestor [
                  ReferTo 0]
                  (length [ReferTo 0]) 0)]>
          [ReferTo 0]) n 0
       (ancestor [ReferTo 0]
          (length [ReferTo 0]) 0))
    (length [ReferTo 0])
s: Tree
step3: false
hS: ReferTo 0 = Ancestor s
False exact step3. }
head, _0: Slot
_1: list Slot
h: withoutCyclesN (head :: _0 :: _1)
  (length (head :: _0 :: _1))
h1: noIllegalIndices (head :: _0 :: _1)
n, u: nat
hU: u < length (head :: _0 :: _1)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x =
nth u (head :: _0 :: _1) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: _0 :: _1)
             (length (head :: _0 :: _1)) u)]>
     (head :: _0 :: _1))
  (length (head :: _0 :: _1))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: _0 :: _1)
                  (length (head :: _0 :: _1)) u)]>
          (head :: _0 :: _1)) n x
       (ancestor (head :: _0 :: _1)
          (length (head :: _0 :: _1)) x))
    (length (head :: _0 :: _1))
step1: u < length (head :: _0 :: _1)
→ x < length (head :: _0 :: _1)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: _0 :: _1)
                 (length (head :: _0 :: _1)) u)]>
         (head :: _0 :: _1))
      (length (head :: _0 :: _1)) x =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
hX1: x < length (head :: _0 :: _1)
s: Tree
hS: nth x (head :: _0 :: _1) (Ancestor Unit) =
Ancestor s
2 ≤ length (head :: _0 :: _1) simpl.
head, _0: Slot
_1: list Slot
h: withoutCyclesN (head :: _0 :: _1)
  (length (head :: _0 :: _1))
h1: noIllegalIndices (head :: _0 :: _1)
n, u: nat
hU: u < length (head :: _0 :: _1)
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x =
nth u (head :: _0 :: _1) (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo
          (ancestor (head :: _0 :: _1)
             (length (head :: _0 :: _1)) u)]>
     (head :: _0 :: _1))
  (length (head :: _0 :: _1))
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor 
                  (head :: _0 :: _1)
                  (length (head :: _0 :: _1)) u)]>
          (head :: _0 :: _1)) n x
       (ancestor (head :: _0 :: _1)
          (length (head :: _0 :: _1)) x))
    (length (head :: _0 :: _1))
step1: u < length (head :: _0 :: _1)
→ x < length (head :: _0 :: _1)
  → ancestor
      (<[u:=ReferTo
              (ancestor (head :: _0 :: _1)
                 (length (head :: _0 :: _1)) u)]>
         (head :: _0 :: _1))
      (length (head :: _0 :: _1)) x =
    ancestor (head :: _0 :: _1)
      (length (head :: _0 :: _1)) x
hX1: x < length (head :: _0 :: _1)
s: Tree
hS: nth x (head :: _0 :: _1) (Ancestor Unit) =
Ancestor s
2 ≤ S (S (length _1)) lia. }
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
    destruct (ltac:(exists (length dsu - 2); lia) : exists n, length dsu = S (S n)) as [nn hn].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
    assert (hg : ancestor dsu (length dsu) u = x).
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
ancestor dsu (length dsu) u = x
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
    {
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
ancestor dsu (length dsu) u = x rewrite hn.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
ancestor dsu (S (S nn)) u = x simpl.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
match nth u dsu (Ancestor Unit) with
| ReferTo _0 =>
    match nth _0 dsu (Ancestor Unit) with
    | ReferTo _1 => ancestor dsu nn _1
    | Ancestor _ => _0
    end
| Ancestor _ => u
end = x rewrite <- hX, hS.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
x = x reflexivity. }
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
withoutCyclesN
  (pathCompress
     (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
     n x (ancestor dsu (length dsu) u)) 
  (length dsu)
    rewrite hg.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
withoutCyclesN
  (pathCompress (<[u:=ReferTo x]> dsu) n x x)
  (length dsu)
    assert (hh : pathCompress (<[u:=ReferTo x]> dsu) n x x = <[u:=ReferTo x]> dsu).
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
pathCompress (<[u:=ReferTo x]> dsu) n x x =
<[u:=ReferTo x]> dsu
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
hh: pathCompress (<[u:=ReferTo x]> dsu) n x x =
<[u:=ReferTo x]> dsu
withoutCyclesN
  (pathCompress (<[u:=ReferTo x]> dsu) n x x)
  (length dsu)
    {
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
pathCompress (<[u:=ReferTo x]> dsu) n x x =
<[u:=ReferTo x]> dsu destruct n.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 0 _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) 0 x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
pathCompress (<[u:=ReferTo x]> dsu) 0 x x =
<[u:=ReferTo x]> dsu
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, u: nat
hU: u < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
pathCompress (<[u:=ReferTo x]> dsu) (S _0) x x =
<[u:=ReferTo x]> dsu {
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 0 _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) 0 x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
pathCompress (<[u:=ReferTo x]> dsu) 0 x x =
<[u:=ReferTo x]> dsu easy. }
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, u: nat
hU: u < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
pathCompress (<[u:=ReferTo x]> dsu) (S _0) x x =
<[u:=ReferTo x]> dsu simpl.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, u: nat
hU: u < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
match
  nth x (<[u:=ReferTo x]> dsu) (Ancestor Unit)
with
| ReferTo _1 =>
    pathCompress
      (<[x:=ReferTo x]> (<[u:=ReferTo x]> dsu)) _0 _1
      x
| Ancestor _ => <[u:=ReferTo x]> dsu
end = <[u:=ReferTo x]> dsu destruct (decide (u = x)) as [hd | hd].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, u: nat
hU: u < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
hd: u = x
match
  nth x (<[u:=ReferTo x]> dsu) (Ancestor Unit)
with
| ReferTo _1 =>
    pathCompress
      (<[x:=ReferTo x]> (<[u:=ReferTo x]> dsu)) _0 _1
      x
| Ancestor _ => <[u:=ReferTo x]> dsu
end = <[u:=ReferTo x]> dsu
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, u: nat
hU: u < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
hd: u ≠ x
match
  nth x (<[u:=ReferTo x]> dsu) (Ancestor Unit)
with
| ReferTo _1 =>
    pathCompress
      (<[x:=ReferTo x]> (<[u:=ReferTo x]> dsu)) _0 _1
      x
| Ancestor _ => <[u:=ReferTo x]> dsu
end = <[u:=ReferTo x]> dsu
      -
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, u: nat
hU: u < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
hd: u = x
match
  nth x (<[u:=ReferTo x]> dsu) (Ancestor Unit)
with
| ReferTo _1 =>
    pathCompress
      (<[x:=ReferTo x]> (<[u:=ReferTo x]> dsu)) _0 _1
      x
| Ancestor _ => <[u:=ReferTo x]> dsu
end = <[u:=ReferTo x]> dsu subst u.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, x: nat
step1: x < length dsu
→ x < length dsu
  → ancestor
      (<[x:=ReferTo
              (ancestor dsu (length dsu) x)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
step: withoutCyclesN
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[x:=ReferTo
               (ancestor dsu (length dsu) x)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
hX: ReferTo x = nth x dsu (Ancestor Unit)
hU: x < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) x = x
match
  nth x (<[x:=ReferTo x]> dsu) (Ancestor Unit)
with
| ReferTo _1 =>
    pathCompress
      (<[x:=ReferTo x]> (<[x:=ReferTo x]> dsu)) _0 _1
      x
| Ancestor _ => <[x:=ReferTo x]> dsu
end = <[x:=ReferTo x]> dsu rewrite hS in hX.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, x: nat
step1: x < length dsu
→ x < length dsu
  → ancestor
      (<[x:=ReferTo
              (ancestor dsu (length dsu) x)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
step: withoutCyclesN
  (<[x:=ReferTo (ancestor dsu (length dsu) x)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[x:=ReferTo
               (ancestor dsu (length dsu) x)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
s: Tree
hX: ReferTo x = Ancestor s
hU: x < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
hX1: x < length dsu
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) x = x
match
  nth x (<[x:=ReferTo x]> dsu) (Ancestor Unit)
with
| ReferTo _1 =>
    pathCompress
      (<[x:=ReferTo x]> (<[x:=ReferTo x]> dsu)) _0 _1
      x
| Ancestor _ => <[x:=ReferTo x]> dsu
end = <[x:=ReferTo x]> dsu easy.
      -
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, u: nat
hU: u < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
hd: u ≠ x
match
  nth x (<[u:=ReferTo x]> dsu) (Ancestor Unit)
with
| ReferTo _1 =>
    pathCompress
      (<[x:=ReferTo x]> (<[u:=ReferTo x]> dsu)) _0 _1
      x
| Ancestor _ => <[u:=ReferTo x]> dsu
end = <[u:=ReferTo x]> dsu rewrite nth_lookup, list_lookup_insert_ne, <- nth_lookup, hS.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, u: nat
hU: u < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
hd: u ≠ x
<[u:=ReferTo x]> dsu = <[u:=ReferTo x]> dsu
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, u: nat
hU: u < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
hd: u ≠ x
u ≠ x {
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, u: nat
hU: u < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
hd: u ≠ x
<[u:=ReferTo x]> dsu = <[u:=ReferTo x]> dsu reflexivity. }
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
_0, u: nat
hU: u < length dsu
IH: ∀ _1 : list Slot,
  withoutCyclesN _1 (length _1)
  → noIllegalIndices _1
    → ∀ _2 : nat,
        _2 < length _1
        → withoutCyclesN
            (pathCompress _1 
               (S _0) _2
               (ancestor _1 (length _1) _2))
            (length _1)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) (S _0) x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
hd: u ≠ x
u ≠ x assumption. }
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
hh: pathCompress (<[u:=ReferTo x]> dsu) n x x =
<[u:=ReferTo x]> dsu
withoutCyclesN
  (pathCompress (<[u:=ReferTo x]> dsu) n x x)
  (length dsu)
    rewrite hh.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
hh: pathCompress (<[u:=ReferTo x]> dsu) n x x =
<[u:=ReferTo x]> dsu
withoutCyclesN (<[u:=ReferTo x]> dsu) (length dsu) clear hh.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
withoutCyclesN (<[u:=ReferTo x]> dsu) (length dsu) intros i hh.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
i: nat
hh: i < length dsu
match
  nth
    (ancestor (<[u:=ReferTo x]> dsu)
       (length (<[u:=ReferTo x]> dsu)) i)
    (<[u:=ReferTo x]> dsu) 
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite <- hg, insert_length, <- ancestorInsert; try assumption; [| now rewrite <- hX].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
i: nat
hh: i < length dsu
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
    destruct (decide (ancestor dsu (length dsu) i = u)) as [hs | hs].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
i: nat
hh: i < length dsu
hs: ancestor dsu (length dsu) i = u
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
i: nat
hh: i < length dsu
hs: ancestor dsu (length dsu) i ≠ u
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
    +
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
i: nat
hh: i < length dsu
hs: ancestor dsu (length dsu) i = u
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end pose proof h i hh as step2.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
i: nat
hh: i < length dsu
hs: ancestor dsu (length dsu) i = u
step2: match
  nth (ancestor dsu (length dsu) i) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite hs, <- hX in step2.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
i: nat
hh: i < length dsu
hs: ancestor dsu (length dsu) i = u
step2: false
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end exfalso.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
i: nat
hh: i < length dsu
hs: ancestor dsu (length dsu) i = u
step2: false
False exact step2.
    +
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
i: nat
hh: i < length dsu
hs: ancestor dsu (length dsu) i ≠ u
match
  nth (ancestor dsu (length dsu) i)
    (<[u:=ReferTo (ancestor dsu (length dsu) u)]> dsu)
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite nth_lookup, list_lookup_insert_ne, <- nth_lookup; [| lia].
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
i: nat
hh: i < length dsu
hs: ancestor dsu (length dsu) i ≠ u
match
  nth (ancestor dsu (length dsu) i) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end pose proof h i hh as step2.
dsu: list Slot
h: withoutCyclesN dsu (length dsu)
h1: noIllegalIndices dsu
n, u: nat
hU: u < length dsu
IH: ∀ _0 : list Slot,
  withoutCyclesN _0 (length _0)
  → noIllegalIndices _0
    → ∀ _1 : nat,
        _1 < length _0
        → withoutCyclesN
            (pathCompress _0 n _1
               (ancestor _0 (length _0) _1))
            (length _0)
x: nat
hX: ReferTo x = nth u dsu (Ancestor Unit)
step: withoutCyclesN
  (<[u:=ReferTo (ancestor dsu (length dsu) u)]>
     dsu) (length dsu)
→ withoutCyclesN
    (pathCompress
       (<[u:=ReferTo
               (ancestor dsu (length dsu) u)]>
          dsu) n x
       (ancestor dsu (length dsu) x))
    (length dsu)
step1: u < length dsu
→ x < length dsu
  → ancestor
      (<[u:=ReferTo
              (ancestor dsu (length dsu) u)]>
         dsu) (length dsu) x =
    ancestor dsu (length dsu) x
hX1: x < length dsu
s: Tree
hS: nth x dsu (Ancestor Unit) = Ancestor s
ht: 2 ≤ length dsu
nn: nat
hn: length dsu = S (S nn)
hg: ancestor dsu (length dsu) u = x
i: nat
hh: i < length dsu
hs: ancestor dsu (length dsu) i ≠ u
step2: match
  nth (ancestor dsu (length dsu) i) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
match
  nth (ancestor dsu (length dsu) i) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end exact step2.
Qed.

Lemma pathCompressPreservesAncestorLength (dsu : list Slot) (h1 : noIllegalIndices dsu) (h2 : withoutCyclesN dsu (length dsu)) (n a b c : nat) (hA : a < length dsu) (hB : b < length dsu) (hC : c < length dsu) (hSameRoot : ancestor dsu (length dsu) c = ancestor dsu (length dsu) a) : ancestor (pathCompress dsu n c (ancestor dsu (length dsu) a)) (length dsu) b = ancestor dsu (length dsu) b.
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
n, a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
ancestor
  (pathCompress dsu n c (ancestor dsu (length dsu) a))
  (length dsu) b = ancestor dsu (length dsu) b
Proof.
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
n, a, b, c: nat
hA: a < length dsu
hB: b < length dsu
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
ancestor
  (pathCompress dsu n c (ancestor dsu (length dsu) a))
  (length dsu) b = ancestor dsu (length dsu) b
  revert dsu h1 h2 hA hB c hC hSameRoot.
n, a, b: nat
∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
  induction n as [| n IH].
a, b: nat
∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 0 _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 
                     (S n) _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b {
a, b: nat
∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 0 _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b easy. }
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 
                     (S n) _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
  intros dsu h1 h2 hA hB c hC hSameRoot.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
ancestor
  (pathCompress dsu (S n) c
     (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b
  rewrite (ltac:(simpl; reflexivity) : pathCompress dsu (S n) _ = _).
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
ancestor
  match nth c dsu (Ancestor Unit) with
  | ReferTo _0 =>
      pathCompress
        (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
           dsu) n _0 (ancestor dsu (length dsu) a)
  | Ancestor _ => dsu
  end (length dsu) b = 
ancestor dsu (length dsu) b
  remember (nth c dsu (Ancestor Unit)) as x eqn:hx.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: Slot
hx: x = nth c dsu (Ancestor Unit)
ancestor
  match x with
  | ReferTo _0 =>
      pathCompress
        (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
           dsu) n _0 (ancestor dsu (length dsu) a)
  | Ancestor _ => dsu
  end (length dsu) b = 
ancestor dsu (length dsu) b destruct x as [x | x]; symmetry in hx; [| reflexivity].
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b
  assert (stepC : match nth c dsu (Ancestor Unit) with | ReferTo _ => true | Ancestor _ => false end).
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b {
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end rewrite hx.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
true easy. }
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b
  pose proof h1 c _ hx as stepX.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b
  pose proof (fun g1 g2 g3 g4 g5 => IH (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu) g1 g2 g3 g4 x g5) as step.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
step: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
→ withoutCyclesN
    (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
       dsu)
    (length
       (<[c:=ReferTo
               (ancestor dsu (length dsu) a)]>
          dsu))
  → a <
    length
      (<[c:=ReferTo
              (ancestor dsu (length dsu) a)]>
         dsu)
    → b <
      length
        (<[c:=ReferTo
                (ancestor dsu (length dsu) a)]>
           dsu)
      → x <
        length
          (<[c:=ReferTo
                  (ancestor dsu (length dsu) a)]>
             dsu)
        → ancestor
            (<[c:=ReferTo
                    (ancestor dsu 
                     (length dsu) a)]> dsu)
            (length
               (<[c:=ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu)) x =
          ancestor
            (<[c:=ReferTo
                    (ancestor dsu 
                     (length dsu) a)]> dsu)
            (length
               (<[c:=ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu)) a
          → ancestor
              (pathCompress
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu) n x
                 (ancestor
                    (<[c:=
                     ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu)
                    (length
                     (<[c:=
                     ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu)) a))
              (length
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu)) b =
            ancestor
              (<[c:=ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu)
              (length
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu)) b
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b rewrite insert_length, <- hSameRoot, <- !ancestorInsert, !hSameRoot in step; try (assumption || lia).
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
step: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
→ withoutCyclesN
    (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
       dsu) (length dsu)
  → a < length dsu
    → b < length dsu
      → x < length dsu
        → ancestor dsu (length dsu) x =
          ancestor dsu (length dsu) a
          → ancestor
              (pathCompress
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu) n x
                 (ancestor dsu (length dsu) a))
              (length dsu) b =
            ancestor dsu (length dsu) b
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b
  pose proof pathCompressPreservesNoIllegalIndices _ h1 1 c (ancestor dsu (length dsu) a) hC ltac:(apply ancestorLtLength; assumption) as stepN.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
step: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
→ withoutCyclesN
    (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
       dsu) (length dsu)
  → a < length dsu
    → b < length dsu
      → x < length dsu
        → ancestor dsu (length dsu) x =
          ancestor dsu (length dsu) a
          → ancestor
              (pathCompress
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu) n x
                 (ancestor dsu (length dsu) a))
              (length dsu) b =
            ancestor dsu (length dsu) b
stepN: noIllegalIndices
  (pathCompress dsu 1 c
     (ancestor dsu (length dsu) a))
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b simpl in stepN.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
step: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
→ withoutCyclesN
    (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
       dsu) (length dsu)
  → a < length dsu
    → b < length dsu
      → x < length dsu
        → ancestor dsu (length dsu) x =
          ancestor dsu (length dsu) a
          → ancestor
              (pathCompress
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu) n x
                 (ancestor dsu (length dsu) a))
              (length dsu) b =
            ancestor dsu (length dsu) b
stepN: noIllegalIndices
  match nth c dsu (Ancestor Unit) with
  | ReferTo _ =>
      <[c:=ReferTo
             (ancestor dsu (length dsu) a)]>
        dsu
  | Ancestor _ => dsu
  end
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b rewrite hx in stepN.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
step: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
→ withoutCyclesN
    (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
       dsu) (length dsu)
  → a < length dsu
    → b < length dsu
      → x < length dsu
        → ancestor dsu (length dsu) x =
          ancestor dsu (length dsu) a
          → ancestor
              (pathCompress
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu) n x
                 (ancestor dsu (length dsu) a))
              (length dsu) b =
            ancestor dsu (length dsu) b
stepN: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b
  pose proof pathCompressPreservesWithoutCycles _ h2 h1 1 c hC as stepW.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
step: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
→ withoutCyclesN
    (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
       dsu) (length dsu)
  → a < length dsu
    → b < length dsu
      → x < length dsu
        → ancestor dsu (length dsu) x =
          ancestor dsu (length dsu) a
          → ancestor
              (pathCompress
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu) n x
                 (ancestor dsu (length dsu) a))
              (length dsu) b =
            ancestor dsu (length dsu) b
stepN: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
stepW: withoutCyclesN
  (pathCompress dsu 1 c
     (ancestor dsu (length dsu) c))
  (length dsu)
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b simpl in stepW.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
step: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
→ withoutCyclesN
    (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
       dsu) (length dsu)
  → a < length dsu
    → b < length dsu
      → x < length dsu
        → ancestor dsu (length dsu) x =
          ancestor dsu (length dsu) a
          → ancestor
              (pathCompress
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu) n x
                 (ancestor dsu (length dsu) a))
              (length dsu) b =
            ancestor dsu (length dsu) b
stepN: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
stepW: withoutCyclesN
  match nth c dsu (Ancestor Unit) with
  | ReferTo _ =>
      <[c:=ReferTo
             (ancestor dsu (length dsu) c)]>
        dsu
  | Ancestor _ => dsu
  end (length dsu)
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b rewrite hx, hSameRoot in stepW.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
step: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
→ withoutCyclesN
    (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
       dsu) (length dsu)
  → a < length dsu
    → b < length dsu
      → x < length dsu
        → ancestor dsu (length dsu) x =
          ancestor dsu (length dsu) a
          → ancestor
              (pathCompress
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu) n x
                 (ancestor dsu (length dsu) a))
              (length dsu) b =
            ancestor dsu (length dsu) b
stepN: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
stepW: withoutCyclesN
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu) (length dsu)
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b
  pose proof ancestorReferTo dsu c x hC h1 h2 hx as stepD.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
step: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
→ withoutCyclesN
    (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
       dsu) (length dsu)
  → a < length dsu
    → b < length dsu
      → x < length dsu
        → ancestor dsu (length dsu) x =
          ancestor dsu (length dsu) a
          → ancestor
              (pathCompress
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu) n x
                 (ancestor dsu (length dsu) a))
              (length dsu) b =
            ancestor dsu (length dsu) b
stepN: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
stepW: withoutCyclesN
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu) (length dsu)
stepD: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) x
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b rewrite hSameRoot in stepD.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
step: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
→ withoutCyclesN
    (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
       dsu) (length dsu)
  → a < length dsu
    → b < length dsu
      → x < length dsu
        → ancestor dsu (length dsu) x =
          ancestor dsu (length dsu) a
          → ancestor
              (pathCompress
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu) n x
                 (ancestor dsu (length dsu) a))
              (length dsu) b =
            ancestor dsu (length dsu) b
stepN: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
stepW: withoutCyclesN
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu) (length dsu)
stepD: ancestor dsu (length dsu) a =
ancestor dsu (length dsu) x
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b symmetry in stepD.
n, a, b: nat
IH: ∀ _0 : list Slot,
  noIllegalIndices _0
  → withoutCyclesN _0 (length _0)
    → a < length _0
      → b < length _0
        → ∀ _1 : nat,
            _1 < length _0
            → ancestor _0 (length _0) _1 =
              ancestor _0 (length _0) a
              → ancestor
                  (pathCompress _0 n _1
                     (ancestor _0 (length _0) a))
                  (length _0) b =
                ancestor _0 (length _0) b
dsu: list Slot
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
hA: a < length dsu
hB: b < length dsu
c: nat
hC: c < length dsu
hSameRoot: ancestor dsu (length dsu) c =
ancestor dsu (length dsu) a
x: nat
hx: nth c dsu (Ancestor Unit) = ReferTo x
stepC: match nth c dsu (Ancestor Unit) with
| ReferTo _ => true
| Ancestor _ => false
end
stepX: x < length dsu
step: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
→ withoutCyclesN
    (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
       dsu) (length dsu)
  → a < length dsu
    → b < length dsu
      → x < length dsu
        → ancestor dsu (length dsu) x =
          ancestor dsu (length dsu) a
          → ancestor
              (pathCompress
                 (<[c:=
                    ReferTo
                     (ancestor dsu 
                     (length dsu) a)]> dsu) n x
                 (ancestor dsu (length dsu) a))
              (length dsu) b =
            ancestor dsu (length dsu) b
stepN: noIllegalIndices
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu)
stepW: withoutCyclesN
  (<[c:=ReferTo (ancestor dsu (length dsu) a)]>
     dsu) (length dsu)
stepD: ancestor dsu (length dsu) x =
ancestor dsu (length dsu) a
ancestor
  (pathCompress
     (<[c:=ReferTo (ancestor dsu (length dsu) a)]> dsu)
     n x (ancestor dsu (length dsu) a)) 
  (length dsu) b = ancestor dsu (length dsu) b
  apply step; assumption.
Qed.

Definition modelScore (interactions : list (Z * Z)) := dsuScore (dsuFromInteractions (repeat (Ancestor Unit) 100) (map (fun (x : Z * Z) => let (a, b) := x in (Z.to_nat a, Z.to_nat b)) interactions)).

Definition getBalanceInvoke (state : BlockchainState) (communication : list Z) :=
  match invokeContract (repeat 1%Z 20) (repeat 0%Z 20) 0 state state communication 1 with
  | Some (a, b) => getBalance (b (repeat 1%Z 20))
  | None => 0%Z
  end.

Lemma initializeArray a b n nextCommands (h : n <= 100) (l : list Z) (hL : length l = 100) : invokeContractAux (repeat 1%Z 20) (repeat 0%Z 20) 0
  state state [a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0) (λ _0 : arrayIndex0,
  match
  _0 as _1
return
  (list
  (arrayType
  arrayIndex0
  environment0
  _1))
with
| arraydef_0__dsu => l
| arraydef_0__hasBeenInitialized =>
    [1%Z]
| arraydef_0__result =>
    [0%Z]
end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
  (λ _ : varsfuncdef_0__main, 0%Z)
  (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
  (λ _ : varsfuncdef_0__main, false)
  (λ _ : varsfuncdef_0__main, 0%Z)
  (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
  (loop n
  (λ _0 : nat,
  dropWithinLoop
  (liftToWithinLoop
  (Done
  (WithLocalVariables arrayIndex0
  (arrayType arrayIndex0
  environment0)
  varsfuncdef_0__main)
  withLocalVariablesReturnValue Z
  (100%Z - Z.of_nat _0 - 1)%Z >>=
λ _1 : Z,
  Done
  (WithLocalVariables arrayIndex0
  (arrayType arrayIndex0
  environment0)
  varsfuncdef_0__main)
  withLocalVariablesReturnValue Z
  (coerceInt (coerceInt (- (1)) 64)
  8) >>=
λ _2 : Z,
  Dispatch
  (WithLocalVariables arrayIndex0
  (arrayType arrayIndex0
  environment0)
  varsfuncdef_0__main)
  withLocalVariablesReturnValue
  (withLocalVariablesReturnValue
  (DoWithArrays arrayIndex0
  (arrayType arrayIndex0
  environment0)
  varsfuncdef_0__main
  (Store arrayIndex0
  (arrayType arrayIndex0
  environment0)
  arraydef_0__dsu _1
  _2)))
  (DoWithArrays arrayIndex0
  (arrayType arrayIndex0
  environment0)
  varsfuncdef_0__main
  (Store arrayIndex0
  (arrayType arrayIndex0
  environment0)
  arraydef_0__dsu _1
  _2))
  (λ _3 : withLocalVariablesReturnValue
  (DoWithArrays
  arrayIndex0
  (arrayType
  arrayIndex0
  environment0)
  varsfuncdef_0__main
  (Store arrayIndex0
  (arrayType
  arrayIndex0
  environment0)
  arraydef_0__dsu
  _1
  _2)),
  Done
  (WithLocalVariables
  arrayIndex0
  (arrayType arrayIndex0
  environment0)
  varsfuncdef_0__main)
  withLocalVariablesReturnValue
  (withLocalVariablesReturnValue
  (DoWithArrays
  arrayIndex0
  (arrayType
  arrayIndex0
  environment0)
  varsfuncdef_0__main
  (Store arrayIndex0
  (arrayType
  arrayIndex0
  environment0)
  arraydef_0__dsu _1
  _2)))
  _3)) >>=
λ _ : (),
  Done
  (WithinLoop arrayIndex0
  (arrayType arrayIndex0
  environment0)
  varsfuncdef_0__main)
  withinLoopReturnValue ()
  ())) >>= (fun _ => nextCommands))) = invokeContractAux (repeat 1%Z 20) (repeat 0%Z 20) 0
  state state [a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0) (λ _0 : arrayIndex0,
  match
  _0 as _1
return
  (list
  (arrayType
  arrayIndex0
  environment0
  _1))
with
| arraydef_0__dsu =>
    take (100 - n) l ++ repeat 255%Z n
| arraydef_0__hasBeenInitialized =>
    [1%Z]
| arraydef_0__result =>
    [0%Z]
end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
  (λ _ : varsfuncdef_0__main, 0%Z)
  (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
  (λ _ : varsfuncdef_0__main, false)
  (λ _ : varsfuncdef_0__main, 0%Z)
  (λ _ : varsfuncdef_0__main, repeat 0%Z 20) nextCommands).
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: n ≤ 100
l: list Z
hL: length l = 100
invokeContractAux (repeat 1%Z 20) (repeat 0%Z 20) 0
  state state [a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                         environment0)
                      varsfuncdef_0__main)
                   withLocalVariablesReturnValue Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                     (WithLocalVariables arrayIndex0
                        (arrayType arrayIndex0
                           environment0)
                        varsfuncdef_0__main)
                     withLocalVariablesReturnValue Z
                     (coerceInt (coerceInt (- (1)) 64)
                        8) >>=
                   λ _2 : Z,
                     Dispatch
                       (WithLocalVariables arrayIndex0
                          (arrayType arrayIndex0
                             environment0)
                          varsfuncdef_0__main)
                       withLocalVariablesReturnValue
                       (withLocalVariablesReturnValue
                          (DoWithArrays arrayIndex0
                             (arrayType arrayIndex0
                                environment0)
                             varsfuncdef_0__main
                             (Store arrayIndex0
                                (arrayType arrayIndex0
                                   environment0)
                                arraydef_0__dsu _1 _2)))
                       (DoWithArrays arrayIndex0
                          (arrayType arrayIndex0
                             environment0)
                          varsfuncdef_0__main
                          (Store arrayIndex0
                             (arrayType arrayIndex0
                                environment0)
                             arraydef_0__dsu _1 _2))
                       (λ _3 : withLocalVariablesReturnValue
                                 (DoWithArrays
                                    arrayIndex0 (...)
                                    varsfuncdef_0__main
                                    (...)),
                          Done
                            (WithLocalVariables
                               arrayIndex0
                               (arrayType arrayIndex0
                                  environment0)
                               varsfuncdef_0__main)
                            withLocalVariablesReturnValue
                            (withLocalVariablesReturnValue
                               (DoWithArrays
                                  arrayIndex0 (...)
                                  varsfuncdef_0__main
                                  (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                     (arrayType arrayIndex0
                        environment0)
                     varsfuncdef_0__main)
                  withinLoopReturnValue () ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) (repeat 0%Z 20) 0
  state state [a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n) l ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands)
Proof.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: n ≤ 100
l: list Z
hL: length l = 100
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                   withLocalVariablesReturnValue Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (coerceInt 
                    (coerceInt (- (1)) 64) 8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _1
                    _2)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _1
                    _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)),
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                  withinLoopReturnValue () ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n) l ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands)
  induction n as [| n IH] in l, hL, h |- *.
a, b: Z
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: 0 ≤ 100
l: list Z
hL: length l = 100
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (loop 0
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                   withLocalVariablesReturnValue Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (coerceInt 
                    (coerceInt (- (1)) 64) 8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _1
                    _2)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _1
                    _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)),
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                  withinLoopReturnValue () ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - 0) l ++ repeat 255%Z 0
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands)
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (loop (S n)
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                   withLocalVariablesReturnValue Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (coerceInt 
                    (coerceInt (- (1)) 64) 8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _1
                    _2)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _1
                    _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)),
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                  withinLoopReturnValue () ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands)
  -
a, b: Z
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: 0 ≤ 100
l: list Z
hL: length l = 100
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (loop 0
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                   withLocalVariablesReturnValue Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (coerceInt 
                    (coerceInt (- (1)) 64) 8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _1
                    _2)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _1
                    _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)),
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                  withinLoopReturnValue () ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - 0) l ++ repeat 255%Z 0
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) unfold loop.
a, b: Z
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: 0 ≤ 100
l: list Z
hL: length l = 100
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (Done
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main)
        withLocalVariablesReturnValue () () >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - 0) l ++ repeat 255%Z 0
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite leftIdentity.
a, b: Z
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: 0 ≤ 100
l: list Z
hL: length l = 100
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - 0) l ++ repeat 255%Z 0
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite (ltac:(easy) : repeat _ 0 = []).
a, b: Z
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: 0 ≤ 100
l: list Z
hL: length l = 100
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => take (100 - 0) l ++ []
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite app_nil_r.
a, b: Z
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: 0 ≤ 100
l: list Z
hL: length l = 100
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => take (100 - 0) l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite (ltac:(lia) : 100 - 0 = 100).
a, b: Z
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: 0 ≤ 100
l: list Z
hL: length l = 100
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => take 100 l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite <- hL, firstn_all.
a, b: Z
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: 0 ≤ 100
l: list Z
hL: length l = 100
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) reflexivity.
  -
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (loop (S n)
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                   withLocalVariablesReturnValue Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (coerceInt 
                    (coerceInt (- (1)) 64) 8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _1
                    _2)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _1
                    _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)),
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                  withinLoopReturnValue () ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite loop_S.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue Z
              (100 - Z.of_nat n - 1)%Z >>=
            λ _0 : Z,
              Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withLocalVariablesReturnValue Z
                (coerceInt (coerceInt (- (1)) 64) 8) >>=
              λ _1 : Z,
                Dispatch
                  (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                  withLocalVariablesReturnValue
                  (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _0
                    _1)))
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _0
                    _1))
                  (λ _2 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _0 _1)),
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _0 _1))) _2)) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main)
             withinLoopReturnValue () ()) >>=
      (λ _0 : LoopOutcome,
         match _0 with
         | KeepGoing =>
             loop n
               (λ _1 : nat,
                  dropWithinLoop
                    (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (coerceInt (coerceInt ... 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (...) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (...))
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...))
                    (λ ...,
                    Done ...
                    withLocalVariablesReturnValue ...
                    _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
         | Stop =>
             Done
               (WithLocalVariables arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__main)
               withLocalVariablesReturnValue () ()
         end) >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite !(ltac:(easy) : (coerceInt (coerceInt (- (1)) 64) 8) = 255%Z).
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue Z
              (100 - Z.of_nat n - 1)%Z >>=
            λ _0 : Z,
              Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withLocalVariablesReturnValue Z 255%Z >>=
              λ _1 : Z,
                Dispatch
                  (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                  withLocalVariablesReturnValue
                  (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _0
                    _1)))
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu _0
                    _1))
                  (λ _2 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _0 _1)),
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _0 _1))) _2)) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main)
             withinLoopReturnValue () ()) >>=
      (λ _0 : LoopOutcome,
         match _0 with
         | KeepGoing =>
             loop n
               (λ _1 : nat,
                  dropWithinLoop
                    (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (...) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (...))
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...))
                    (λ ...,
                    Done ...
                    withLocalVariablesReturnValue ...
                    _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
         | Stop =>
             Done
               (WithLocalVariables arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__main)
               withLocalVariablesReturnValue () ()
         end) >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite !leftIdentity.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Dispatch
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue
              (withLocalVariablesReturnValue
                 (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)))
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z))
              (λ _0 : 
                  withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)),
                 Done
                   (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z))) _0)) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main)
             withinLoopReturnValue () ()) >>=
      (λ _0 : LoopOutcome,
         match _0 with
         | KeepGoing =>
             loop n
               (λ _1 : nat,
                  dropWithinLoop
                    (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (...) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (...))
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...))
                    (λ ...,
                    Done ...
                    withLocalVariablesReturnValue ...
                    _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
         | Stop =>
             Done
               (WithLocalVariables arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__main)
               withLocalVariablesReturnValue () ()
         end) >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) unfold liftToWithinLoop at 1.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (dropWithinLoop
        (Dispatch
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue ()
           (DoWithLocalVariables arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)))
           (λ _0 : withinLoopReturnValue
                    (DoWithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z))),
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withinLoopReturnValue () _0) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main)
             withinLoopReturnValue () ()) >>=
      (λ _0 : LoopOutcome,
         match _0 with
         | KeepGoing =>
             loop n
               (λ _1 : nat,
                  dropWithinLoop
                    (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (...) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (...))
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...))
                    (λ ...,
                    Done ...
                    withLocalVariablesReturnValue ...
                    _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
         | Stop =>
             Done
               (WithLocalVariables arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__main)
               withLocalVariablesReturnValue () ()
         end) >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite <- !bindAssoc.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (dropWithinLoop
        (Dispatch
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue ()
           (DoWithLocalVariables arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)))
           (λ _0 : withinLoopReturnValue
                    (DoWithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z))),
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withinLoopReturnValue () _0) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main)
             withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 ...
                    varsfuncdef_0__main ...))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 ...
                    arraydef_0__dsu _2 _3))
                    (λ _4 : ...,
                    Done 
                    (...)
                    withLocalVariablesReturnValue
                    (...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue () ()
        end >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) pose proof dropWithinLoop_2' _ _ _ (DoWithArrays arrayIndex0 (arrayType arrayIndex0 environment0) varsfuncdef_0__main (Store arrayIndex0 (arrayType arrayIndex0 environment0) arraydef_0__dsu (100 - Z.of_nat n - 1) 255%Z)) as step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
step: ∀ _0 : withinLoopReturnValue
         (DoWithLocalVariables arrayIndex0
            (arrayType arrayIndex0 environment0)
            varsfuncdef_0__main
            (DoWithArrays arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main
               (Store arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  arraydef_0__dsu
                  (100 - Z.of_nat n - 1) 255%Z)))
       → Action
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue (),
  dropWithinLoop
    (Dispatch
       (WithinLoop arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withinLoopReturnValue
       (withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))))
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z)))
       (λ _1 : withinLoopReturnValue
                 (DoWithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z))),
          Done
            (WithinLoop arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main)
            withinLoopReturnValue
            (withinLoopReturnValue
               (DoWithLocalVariables
                  arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)))) _1) >>= _0) =
  Dispatch
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main)
    withLocalVariablesReturnValue LoopOutcome
    (DoWithArrays arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main
       (Store arrayIndex0
          (arrayType arrayIndex0 environment0)
          arraydef_0__dsu
          (100 - Z.of_nat n - 1) 255%Z))
    (λ _1 : withLocalVariablesReturnValue
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
       dropWithinLoop (_0 _1))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (dropWithinLoop
        (Dispatch
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue ()
           (DoWithLocalVariables arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)))
           (λ _0 : withinLoopReturnValue
                    (DoWithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z))),
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withinLoopReturnValue () _0) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main)
             withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 ...
                    varsfuncdef_0__main ...))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 ...
                    arraydef_0__dsu _2 _3))
                    (λ _4 : ...,
                    Done 
                    (...)
                    withLocalVariablesReturnValue
                    (...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue () ()
        end >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) assert (h1 : (λ _1 : withinLoopReturnValue
  (DoWithLocalVariables arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__main
  (DoWithArrays arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__main
  (Store arrayIndex0
  (arrayType arrayIndex0 environment0)
  arraydef_0__dsu
  (100 - Z.of_nat n - 1)
  255%Z))),
  Done
  (WithinLoop arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__main)
  withinLoopReturnValue
  (withinLoopReturnValue
  (DoWithLocalVariables arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__main
  (DoWithArrays arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__main
  (Store arrayIndex0
  (arrayType arrayIndex0 environment0)
  arraydef_0__dsu
  (100 - Z.of_nat n - 1) 255%Z))))
  _1) =  (λ _1 : unit,
  Done
  (WithinLoop arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__main)
  withinLoopReturnValue
  (withinLoopReturnValue
  (DoWithLocalVariables arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__main
  (DoWithArrays arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__main
  (Store arrayIndex0
  (arrayType arrayIndex0 environment0)
  arraydef_0__dsu
  (100 - Z.of_nat n - 1) 255%Z))))
  _1) ).
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
step: ∀ _0 : withinLoopReturnValue
         (DoWithLocalVariables arrayIndex0
            (arrayType arrayIndex0 environment0)
            varsfuncdef_0__main
            (DoWithArrays arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main
               (Store arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  arraydef_0__dsu
                  (100 - Z.of_nat n - 1) 255%Z)))
       → Action
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue (),
  dropWithinLoop
    (Dispatch
       (WithinLoop arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withinLoopReturnValue
       (withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))))
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z)))
       (λ _1 : withinLoopReturnValue
                 (DoWithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z))),
          Done
            (WithinLoop arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main)
            withinLoopReturnValue
            (withinLoopReturnValue
               (DoWithLocalVariables
                  arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)))) _1) >>= _0) =
  Dispatch
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main)
    withLocalVariablesReturnValue LoopOutcome
    (DoWithArrays arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main
       (Store arrayIndex0
          (arrayType arrayIndex0 environment0)
          arraydef_0__dsu
          (100 - Z.of_nat n - 1) 255%Z))
    (λ _1 : withLocalVariablesReturnValue
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
       dropWithinLoop (_0 _1))
(λ _0 : withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))),
   Done
     (WithinLoop arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__main) withinLoopReturnValue
     (withinLoopReturnValue
        (DoWithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main
           (DoWithArrays arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (Store arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 arraydef_0__dsu
                 (100 - Z.of_nat n - 1) 255%Z)))) _0) =
λ _0 : (),
  Done
    (WithinLoop arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main) withinLoopReturnValue
    (withinLoopReturnValue
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0 environment0)
                arraydef_0__dsu 
                (100 - Z.of_nat n - 1) 255%Z)))) _0
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
step: ∀ _0 : withinLoopReturnValue
         (DoWithLocalVariables arrayIndex0
            (arrayType arrayIndex0 environment0)
            varsfuncdef_0__main
            (DoWithArrays arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main
               (Store arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  arraydef_0__dsu
                  (100 - Z.of_nat n - 1) 255%Z)))
       → Action
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue (),
  dropWithinLoop
    (Dispatch
       (WithinLoop arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withinLoopReturnValue
       (withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))))
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z)))
       (λ _1 : withinLoopReturnValue
                 (DoWithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z))),
          Done
            (WithinLoop arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main)
            withinLoopReturnValue
            (withinLoopReturnValue
               (DoWithLocalVariables
                  arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)))) _1) >>= _0) =
  Dispatch
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main)
    withLocalVariablesReturnValue LoopOutcome
    (DoWithArrays arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main
       (Store arrayIndex0
          (arrayType arrayIndex0 environment0)
          arraydef_0__dsu
          (100 - Z.of_nat n - 1) 255%Z))
    (λ _1 : withLocalVariablesReturnValue
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
       dropWithinLoop (_0 _1))
h1: (λ _0 : withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))),
   Done
     (WithinLoop arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__main)
     withinLoopReturnValue
     (withinLoopReturnValue
        (DoWithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main
           (DoWithArrays arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (Store arrayIndex0
                 (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                 (100 - Z.of_nat n - 1) 255%Z))))
     _0) =
λ _0 : (),
  Done
    (WithinLoop arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main) withinLoopReturnValue
    (withinLoopReturnValue
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0) arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z))))
    _0
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (dropWithinLoop
        (Dispatch
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue ()
           (DoWithLocalVariables arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)))
           (λ _0 : withinLoopReturnValue
                    (DoWithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z))),
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withinLoopReturnValue () _0) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main)
             withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 ...
                    varsfuncdef_0__main ...))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 ...
                    arraydef_0__dsu _2 _3))
                    (λ _4 : ...,
                    Done 
                    (...)
                    withLocalVariablesReturnValue
                    (...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue () ()
        end >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) {
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
step: ∀ _0 : withinLoopReturnValue
         (DoWithLocalVariables arrayIndex0
            (arrayType arrayIndex0 environment0)
            varsfuncdef_0__main
            (DoWithArrays arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main
               (Store arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  arraydef_0__dsu
                  (100 - Z.of_nat n - 1) 255%Z)))
       → Action
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue (),
  dropWithinLoop
    (Dispatch
       (WithinLoop arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withinLoopReturnValue
       (withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))))
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z)))
       (λ _1 : withinLoopReturnValue
                 (DoWithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z))),
          Done
            (WithinLoop arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main)
            withinLoopReturnValue
            (withinLoopReturnValue
               (DoWithLocalVariables
                  arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)))) _1) >>= _0) =
  Dispatch
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main)
    withLocalVariablesReturnValue LoopOutcome
    (DoWithArrays arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main
       (Store arrayIndex0
          (arrayType arrayIndex0 environment0)
          arraydef_0__dsu
          (100 - Z.of_nat n - 1) 255%Z))
    (λ _1 : withLocalVariablesReturnValue
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
       dropWithinLoop (_0 _1))
(λ _0 : withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))),
   Done
     (WithinLoop arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__main) withinLoopReturnValue
     (withinLoopReturnValue
        (DoWithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main
           (DoWithArrays arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (Store arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 arraydef_0__dsu
                 (100 - Z.of_nat n - 1) 255%Z)))) _0) =
λ _0 : (),
  Done
    (WithinLoop arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main) withinLoopReturnValue
    (withinLoopReturnValue
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0 environment0)
                arraydef_0__dsu 
                (100 - Z.of_nat n - 1) 255%Z)))) _0 apply functional_extensionality_dep.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
step: ∀ _0 : withinLoopReturnValue
         (DoWithLocalVariables arrayIndex0
            (arrayType arrayIndex0 environment0)
            varsfuncdef_0__main
            (DoWithArrays arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main
               (Store arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  arraydef_0__dsu
                  (100 - Z.of_nat n - 1) 255%Z)))
       → Action
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue (),
  dropWithinLoop
    (Dispatch
       (WithinLoop arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withinLoopReturnValue
       (withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))))
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z)))
       (λ _1 : withinLoopReturnValue
                 (DoWithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z))),
          Done
            (WithinLoop arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main)
            withinLoopReturnValue
            (withinLoopReturnValue
               (DoWithLocalVariables
                  arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)))) _1) >>= _0) =
  Dispatch
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main)
    withLocalVariablesReturnValue LoopOutcome
    (DoWithArrays arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main
       (Store arrayIndex0
          (arrayType arrayIndex0 environment0)
          arraydef_0__dsu
          (100 - Z.of_nat n - 1) 255%Z))
    (λ _1 : withLocalVariablesReturnValue
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
       dropWithinLoop (_0 _1))
∀ _0 : withinLoopReturnValue
         (DoWithLocalVariables arrayIndex0
            (arrayType arrayIndex0 environment0)
            varsfuncdef_0__main
            (DoWithArrays arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__main
               (Store arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  arraydef_0__dsu
                  (100 - Z.of_nat n - 1) 255%Z))),
  Done
    (WithinLoop arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main) withinLoopReturnValue
    (withinLoopReturnValue
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0 environment0)
                arraydef_0__dsu 
                (100 - Z.of_nat n - 1) 255%Z)))) _0 =
  Done
    (WithinLoop arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main) withinLoopReturnValue
    (withinLoopReturnValue
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0 environment0)
                arraydef_0__dsu 
                (100 - Z.of_nat n - 1) 255%Z)))) _0 intro x.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
step: ∀ _0 : withinLoopReturnValue
         (DoWithLocalVariables arrayIndex0
            (arrayType arrayIndex0 environment0)
            varsfuncdef_0__main
            (DoWithArrays arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main
               (Store arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  arraydef_0__dsu
                  (100 - Z.of_nat n - 1) 255%Z)))
       → Action
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue (),
  dropWithinLoop
    (Dispatch
       (WithinLoop arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withinLoopReturnValue
       (withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))))
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z)))
       (λ _1 : withinLoopReturnValue
                 (DoWithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z))),
          Done
            (WithinLoop arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main)
            withinLoopReturnValue
            (withinLoopReturnValue
               (DoWithLocalVariables
                  arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)))) _1) >>= _0) =
  Dispatch
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main)
    withLocalVariablesReturnValue LoopOutcome
    (DoWithArrays arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main
       (Store arrayIndex0
          (arrayType arrayIndex0 environment0)
          arraydef_0__dsu
          (100 - Z.of_nat n - 1) 255%Z))
    (λ _1 : withLocalVariablesReturnValue
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
       dropWithinLoop (_0 _1))
x: withinLoopReturnValue
  (DoWithLocalVariables arrayIndex0
     (arrayType arrayIndex0 environment0)
     varsfuncdef_0__main
     (DoWithArrays arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__main
        (Store arrayIndex0
           (arrayType arrayIndex0 environment0)
           arraydef_0__dsu 
           (100 - Z.of_nat n - 1) 255%Z)))
Done
  (WithinLoop arrayIndex0
     (arrayType arrayIndex0 environment0)
     varsfuncdef_0__main) withinLoopReturnValue
  (withinLoopReturnValue
     (DoWithLocalVariables arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__main
        (DoWithArrays arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main
           (Store arrayIndex0
              (arrayType arrayIndex0 environment0)
              arraydef_0__dsu 
              (100 - Z.of_nat n - 1) 255%Z)))) x =
Done
  (WithinLoop arrayIndex0
     (arrayType arrayIndex0 environment0)
     varsfuncdef_0__main) withinLoopReturnValue
  (withinLoopReturnValue
     (DoWithLocalVariables arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__main
        (DoWithArrays arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main
           (Store arrayIndex0
              (arrayType arrayIndex0 environment0)
              arraydef_0__dsu 
              (100 - Z.of_nat n - 1) 255%Z)))) x reflexivity. }
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
step: ∀ _0 : withinLoopReturnValue
         (DoWithLocalVariables arrayIndex0
            (arrayType arrayIndex0 environment0)
            varsfuncdef_0__main
            (DoWithArrays arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main
               (Store arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  arraydef_0__dsu
                  (100 - Z.of_nat n - 1) 255%Z)))
       → Action
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue (),
  dropWithinLoop
    (Dispatch
       (WithinLoop arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withinLoopReturnValue
       (withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))))
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z)))
       (λ _1 : withinLoopReturnValue
                 (DoWithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z))),
          Done
            (WithinLoop arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main)
            withinLoopReturnValue
            (withinLoopReturnValue
               (DoWithLocalVariables
                  arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)))) _1) >>= _0) =
  Dispatch
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main)
    withLocalVariablesReturnValue LoopOutcome
    (DoWithArrays arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main
       (Store arrayIndex0
          (arrayType arrayIndex0 environment0)
          arraydef_0__dsu
          (100 - Z.of_nat n - 1) 255%Z))
    (λ _1 : withLocalVariablesReturnValue
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
       dropWithinLoop (_0 _1))
h1: (λ _0 : withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))),
   Done
     (WithinLoop arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__main)
     withinLoopReturnValue
     (withinLoopReturnValue
        (DoWithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main
           (DoWithArrays arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (Store arrayIndex0
                 (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                 (100 - Z.of_nat n - 1) 255%Z))))
     _0) =
λ _0 : (),
  Done
    (WithinLoop arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main) withinLoopReturnValue
    (withinLoopReturnValue
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0) arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z))))
    _0
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (dropWithinLoop
        (Dispatch
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue ()
           (DoWithLocalVariables arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)))
           (λ _0 : withinLoopReturnValue
                    (DoWithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z))),
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withinLoopReturnValue () _0) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main)
             withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 ...
                    varsfuncdef_0__main ...))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 ...
                    arraydef_0__dsu _2 _3))
                    (λ _4 : ...,
                    Done 
                    (...)
                    withLocalVariablesReturnValue
                    (...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue () ()
        end >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite h1 in step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
step: ∀ _0 : withinLoopReturnValue
         (DoWithLocalVariables arrayIndex0
            (arrayType arrayIndex0 environment0)
            varsfuncdef_0__main
            (DoWithArrays arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main
               (Store arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  arraydef_0__dsu
                  (100 - Z.of_nat n - 1) 255%Z)))
       → Action
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue (),
  dropWithinLoop
    (Dispatch
       (WithinLoop arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withinLoopReturnValue
       (withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))))
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z)))
       (λ _1 : (),
          Done
            (WithinLoop arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main)
            withinLoopReturnValue
            (withinLoopReturnValue
               (DoWithLocalVariables
                  arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)))) _1) >>= _0) =
  Dispatch
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main)
    withLocalVariablesReturnValue LoopOutcome
    (DoWithArrays arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main
       (Store arrayIndex0
          (arrayType arrayIndex0 environment0)
          arraydef_0__dsu
          (100 - Z.of_nat n - 1) 255%Z))
    (λ _1 : withLocalVariablesReturnValue
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
       dropWithinLoop (_0 _1))
h1: (λ _0 : withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))),
   Done
     (WithinLoop arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__main)
     withinLoopReturnValue
     (withinLoopReturnValue
        (DoWithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main
           (DoWithArrays arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (Store arrayIndex0
                 (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                 (100 - Z.of_nat n - 1) 255%Z))))
     _0) =
λ _0 : (),
  Done
    (WithinLoop arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main) withinLoopReturnValue
    (withinLoopReturnValue
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0 environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0) arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z))))
    _0
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (dropWithinLoop
        (Dispatch
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue ()
           (DoWithLocalVariables arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)))
           (λ _0 : withinLoopReturnValue
                    (DoWithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z))),
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withinLoopReturnValue () _0) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main)
             withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 ...
                    varsfuncdef_0__main ...))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 ...
                    arraydef_0__dsu _2 _3))
                    (λ _4 : ...,
                    Done 
                    (...)
                    withLocalVariablesReturnValue
                    (...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue () ()
        end >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) clear h1.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
step: ∀ _0 : withinLoopReturnValue
         (DoWithLocalVariables arrayIndex0
            (arrayType arrayIndex0 environment0)
            varsfuncdef_0__main
            (DoWithArrays arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main
               (Store arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  arraydef_0__dsu
                  (100 - Z.of_nat n - 1) 255%Z)))
       → Action
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue (),
  dropWithinLoop
    (Dispatch
       (WithinLoop arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withinLoopReturnValue
       (withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))))
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z)))
       (λ _1 : (),
          Done
            (WithinLoop arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main)
            withinLoopReturnValue
            (withinLoopReturnValue
               (DoWithLocalVariables
                  arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)))) _1) >>= _0) =
  Dispatch
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main)
    withLocalVariablesReturnValue LoopOutcome
    (DoWithArrays arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main
       (Store arrayIndex0
          (arrayType arrayIndex0 environment0)
          arraydef_0__dsu
          (100 - Z.of_nat n - 1) 255%Z))
    (λ _1 : withLocalVariablesReturnValue
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
       dropWithinLoop (_0 _1))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (dropWithinLoop
        (Dispatch
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue ()
           (DoWithLocalVariables arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__main
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)))
           (λ _0 : withinLoopReturnValue
                    (DoWithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z))),
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withinLoopReturnValue () _0) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__main)
             withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 ...
                    varsfuncdef_0__main ...))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 ...
                    arraydef_0__dsu _2 _3))
                    (λ _4 : ...,
                    Done 
                    (...)
                    withLocalVariablesReturnValue
                    (...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue () ()
        end >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
step: ∀ _0 : withinLoopReturnValue
         (DoWithLocalVariables arrayIndex0
            (arrayType arrayIndex0 environment0)
            varsfuncdef_0__main
            (DoWithArrays arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main
               (Store arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  arraydef_0__dsu
                  (100 - Z.of_nat n - 1) 255%Z)))
       → Action
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withinLoopReturnValue (),
  dropWithinLoop
    (Dispatch
       (WithinLoop arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withinLoopReturnValue
       (withinLoopReturnValue
          (DoWithLocalVariables arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (DoWithArrays arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                varsfuncdef_0__main
                (Store arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   arraydef_0__dsu
                   (100 - Z.of_nat n - 1) 255%Z))))
       (DoWithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__main
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z)))
       (λ _1 : (),
          Done
            (WithinLoop arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__main)
            withinLoopReturnValue
            (withinLoopReturnValue
               (DoWithLocalVariables
                  arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main
                  (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)))) _1) >>= _0) =
  Dispatch
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main)
    withLocalVariablesReturnValue LoopOutcome
    (DoWithArrays arrayIndex0
       (arrayType arrayIndex0 environment0)
       varsfuncdef_0__main
       (Store arrayIndex0
          (arrayType arrayIndex0 environment0)
          arraydef_0__dsu
          (100 - Z.of_nat n - 1) 255%Z))
    (λ _1 : withLocalVariablesReturnValue
              (DoWithArrays arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 varsfuncdef_0__main
                 (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
       dropWithinLoop (_0 _1))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (Dispatch
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main)
        withLocalVariablesReturnValue LoopOutcome
        (DoWithArrays arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main
           (Store arrayIndex0
              (arrayType arrayIndex0 environment0)
              arraydef_0__dsu 
              (100 - Z.of_nat n - 1) 255%Z))
        (λ _ : withLocalVariablesReturnValue
                 (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)),
           dropWithinLoop
             (Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withinLoopReturnValue () ())) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 ...
                    varsfuncdef_0__main ...))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 ...
                    arraydef_0__dsu _2 _3))
                    (λ _4 : ...,
                    Done 
                    (...)
                    withLocalVariablesReturnValue
                    (...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue () ()
        end >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) clear step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (Dispatch
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main)
        withLocalVariablesReturnValue LoopOutcome
        (DoWithArrays arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main
           (Store arrayIndex0
              (arrayType arrayIndex0 environment0)
              arraydef_0__dsu 
              (100 - Z.of_nat n - 1) 255%Z))
        (λ _ : withLocalVariablesReturnValue
                 (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)),
           dropWithinLoop
             (Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withinLoopReturnValue () ())) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 ...
                    varsfuncdef_0__main ...))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 ...
                    arraydef_0__dsu _2 _3))
                    (λ _4 : ...,
                    Done 
                    (...)
                    withLocalVariablesReturnValue
                    (...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue () ()
        end >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) pose proof pushDispatch3 (λ _ : varsfuncdef_0__main, false) (λ _ : varsfuncdef_0__main, 0%Z) (λ _ : varsfuncdef_0__main, repeat 0%Z 20) (Store arrayIndex0 (arrayType arrayIndex0
  environment0)
  arraydef_0__dsu (100 - Z.of_nat n - 1)
  255%Z) as well.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
well: ∀ _0 : LoopOutcome
       → Action
           (WithLocalVariables arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withLocalVariablesReturnValue (),
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__main, false)
    (λ _ : varsfuncdef_0__main, 0%Z)
    (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
    (Dispatch
       (WithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withLocalVariablesReturnValue
       LoopOutcome
       (DoWithArrays arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (Store arrayIndex0
             (arrayType arrayIndex0
                environment0) arraydef_0__dsu
             (100 - Z.of_nat n - 1) 255%Z))
       (λ _ : withLocalVariablesReturnValue
                (DoWithArrays arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   varsfuncdef_0__main
                   (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
          dropWithinLoop
            (Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main)
               withinLoopReturnValue () ())) >>=
     _0) =
  Dispatch
    (WithArrays arrayIndex0
       (arrayType arrayIndex0 environment0))
    withArraysReturnValue ()
    (Store arrayIndex0
       (arrayType arrayIndex0 environment0)
       arraydef_0__dsu 
       (100 - Z.of_nat n - 1) 255%Z)
    (λ _ : withArraysReturnValue
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z),
       eliminateLocalVariables
         (λ _ : varsfuncdef_0__main, false)
         (λ _ : varsfuncdef_0__main, 0%Z)
         (λ _ : varsfuncdef_0__main,
            repeat 0%Z 20) 
         (_0 KeepGoing))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     (Dispatch
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main)
        withLocalVariablesReturnValue LoopOutcome
        (DoWithArrays arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__main
           (Store arrayIndex0
              (arrayType arrayIndex0 environment0)
              arraydef_0__dsu 
              (100 - Z.of_nat n - 1) 255%Z))
        (λ _ : withLocalVariablesReturnValue
                 (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0) arraydef_0__dsu
                    (100 - Z.of_nat n - 1) 255%Z)),
           dropWithinLoop
             (Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__main)
                withinLoopReturnValue () ())) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 ...
                    varsfuncdef_0__main ...))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 ...
                    arraydef_0__dsu _2 _3))
                    (λ _4 : ...,
                    Done 
                    (...)
                    withLocalVariablesReturnValue
                    (...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__main)
              withLocalVariablesReturnValue () ()
        end >>= λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite well.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
well: ∀ _0 : LoopOutcome
       → Action
           (WithLocalVariables arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__main)
           withLocalVariablesReturnValue (),
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__main, false)
    (λ _ : varsfuncdef_0__main, 0%Z)
    (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
    (Dispatch
       (WithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main)
       withLocalVariablesReturnValue
       LoopOutcome
       (DoWithArrays arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__main
          (Store arrayIndex0
             (arrayType arrayIndex0
                environment0) arraydef_0__dsu
             (100 - Z.of_nat n - 1) 255%Z))
       (λ _ : withLocalVariablesReturnValue
                (DoWithArrays arrayIndex0
                   (arrayType arrayIndex0
                    environment0)
                   varsfuncdef_0__main
                   (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu
                    (100 - Z.of_nat n - 1)
                    255%Z)),
          dropWithinLoop
            (Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0
                    environment0)
                  varsfuncdef_0__main)
               withinLoopReturnValue () ())) >>=
     _0) =
  Dispatch
    (WithArrays arrayIndex0
       (arrayType arrayIndex0 environment0))
    withArraysReturnValue ()
    (Store arrayIndex0
       (arrayType arrayIndex0 environment0)
       arraydef_0__dsu 
       (100 - Z.of_nat n - 1) 255%Z)
    (λ _ : withArraysReturnValue
             (Store arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu
                (100 - Z.of_nat n - 1) 255%Z),
       eliminateLocalVariables
         (λ _ : varsfuncdef_0__main, false)
         (λ _ : varsfuncdef_0__main, 0%Z)
         (λ _ : varsfuncdef_0__main,
            repeat 0%Z 20) 
         (_0 KeepGoing))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (Dispatch
     (WithArrays arrayIndex0
        (arrayType arrayIndex0 environment0))
     withArraysReturnValue ()
     (Store arrayIndex0
        (arrayType arrayIndex0 environment0)
        arraydef_0__dsu 
        (100 - Z.of_nat n - 1) 255%Z)
     (λ _ : withArraysReturnValue
              (Store arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 arraydef_0__dsu
                 (100 - Z.of_nat n - 1) 255%Z),
        eliminateLocalVariables
          (λ _ : varsfuncdef_0__main, false)
          (λ _ : varsfuncdef_0__main, 0%Z)
          (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
          (loop n
             (λ _0 : nat,
                dropWithinLoop
                  (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _0 - 1)%Z >>=
                    λ _1 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue ...,
                    Done
                    (WithLocalVariables arrayIndex0
                    ... varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue ...)
                    _3)) >>=
                   λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
           λ _ : (), nextCommands))) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) clear well.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => l
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (Dispatch
     (WithArrays arrayIndex0
        (arrayType arrayIndex0 environment0))
     withArraysReturnValue ()
     (Store arrayIndex0
        (arrayType arrayIndex0 environment0)
        arraydef_0__dsu 
        (100 - Z.of_nat n - 1) 255%Z)
     (λ _ : withArraysReturnValue
              (Store arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 arraydef_0__dsu
                 (100 - Z.of_nat n - 1) 255%Z),
        eliminateLocalVariables
          (λ _ : varsfuncdef_0__main, false)
          (λ _ : varsfuncdef_0__main, 0%Z)
          (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
          (loop n
             (λ _0 : nat,
                dropWithinLoop
                  (liftToWithinLoop
                    (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _0 - 1)%Z >>=
                    λ _1 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue ...,
                    Done
                    (WithLocalVariables arrayIndex0
                    ... varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue ...)
                    _3)) >>=
                   λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
           λ _ : (), nextCommands))) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) autorewrite with advance_program.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
(if
  decide
    (Nat.lt (Z.to_nat (100 - Z.of_nat n - 1))
       (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match decide (_0 = arraydef_0__dsu) with
       | left _1 =>
           eq_rect_r
             (λ _2 : arrayIndex0,
                list
                  (arrayType arrayIndex0 environment0
                    _2))
             (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
                l) _1
       | right _ =>
           match
             _0 as _2
             return
               (list
                  (arrayType arrayIndex0 environment0
                    _2))
           with
           | arraydef_0__dsu => l
           | arraydef_0__hasBeenInitialized => [1%Z]
           | arraydef_0__result => [0%Z]
           end
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       (loop n
          (λ _0 : nat,
             dropWithinLoop
               (liftToWithinLoop
                  (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _0 - 1)%Z >>=
                   λ _1 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue ...,
                    Done
                    (WithLocalVariables arrayIndex0
                    ... varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue ...)
                    _3)) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
        λ _ : (), nextCommands))
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands)
  pose proof IH ltac:(lia) (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]> l) ltac:(now rewrite insert_length) as previous.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
IH: n ≤ 100
→ ∀ _0 : list Z,
    length _0 = 100
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu => _0
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20)
           (loop n
              (λ _1 : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                    (Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z (100 - Z.of_nat _1 - 1)%Z >>=
                    λ _2 : Z,
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64) 8) >>=
                    λ _3 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0
                    (...) varsfuncdef_0__main
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _2 _3))
                    (λ _4 : 
                    withLocalVariablesReturnValue
                    ...,
                    Done
                    (WithLocalVariables
                    arrayIndex0 ...
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    ...) _4)) >>=
                    λ _ : (),
                    Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
            λ _ : (), nextCommands)) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state state [
        a; b] 1 arrayIndex0
        arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0
                    environment0 _2))
           with
           | arraydef_0__dsu =>
               take (100 - n) _0 ++
               repeat 255%Z n
           | arraydef_0__hasBeenInitialized =>
               [1%Z]
           | arraydef_0__result => [0%Z]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main,
              repeat 0%Z 20) nextCommands)
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64)
                    8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(if
  decide
    (Nat.lt (Z.to_nat (100 - Z.of_nat n - 1))
       (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match decide (_0 = arraydef_0__dsu) with
       | left _1 =>
           eq_rect_r
             (λ _2 : arrayIndex0,
                list
                  (arrayType arrayIndex0 environment0
                    _2))
             (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
                l) _1
       | right _ =>
           match
             _0 as _2
             return
               (list
                  (arrayType arrayIndex0 environment0
                    _2))
           with
           | arraydef_0__dsu => l
           | arraydef_0__hasBeenInitialized => [1%Z]
           | arraydef_0__result => [0%Z]
           end
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       (loop n
          (λ _0 : nat,
             dropWithinLoop
               (liftToWithinLoop
                  (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _0 - 1)%Z >>=
                   λ _1 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue ...,
                    Done
                    (WithLocalVariables arrayIndex0
                    ... varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue ...)
                    _3)) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
        λ _ : (), nextCommands))
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) clear IH.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64)
                    8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(if
  decide
    (Nat.lt (Z.to_nat (100 - Z.of_nat n - 1))
       (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match decide (_0 = arraydef_0__dsu) with
       | left _1 =>
           eq_rect_r
             (λ _2 : arrayIndex0,
                list
                  (arrayType arrayIndex0 environment0
                    _2))
             (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
                l) _1
       | right _ =>
           match
             _0 as _2
             return
               (list
                  (arrayType arrayIndex0 environment0
                    _2))
           with
           | arraydef_0__dsu => l
           | arraydef_0__hasBeenInitialized => [1%Z]
           | arraydef_0__result => [0%Z]
           end
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       (loop n
          (λ _0 : nat,
             dropWithinLoop
               (liftToWithinLoop
                  (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _0 - 1)%Z >>=
                   λ _1 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue ...,
                    Done
                    (WithLocalVariables arrayIndex0
                    ... varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue ...)
                    _3)) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
        λ _ : (), nextCommands))
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands)

  assert (hh : (λ _0 : arrayIndex0,
  match
  _0 as _1
return
  (list
  (arrayType
  arrayIndex0
  environment0
  _1))
with
| arraydef_0__dsu =>
    <[Z.to_nat
  (100 -
Z.of_nat n -
1):=255%Z]>
  l
| arraydef_0__hasBeenInitialized =>
    [1%Z]
| arraydef_0__result =>
    [0%Z]
end) = (λ _0 : arrayIndex0,
  match
  decide
  (_0 = arraydef_0__dsu)
with
| left _1 =>
  @eq_rect_r arrayIndex0 arraydef_0__dsu
  (fun _2 : arrayIndex0 =>
list (arrayType arrayIndex0 environment0 _2))
  (@insert nat
  (arrayType arrayIndex0 environment0
  arraydef_0__dsu)
  (list
  (arrayType arrayIndex0 environment0
  arraydef_0__dsu))
  (@list_insert
  (arrayType arrayIndex0 environment0
  arraydef_0__dsu))
  (Z.to_nat (Z.sub (Z.sub 100 (Z.of_nat n)) 1))
  255%Z
  l)
  _0
  _1
| right _ =>
    match
  _0 as _2
return
  (list
  (arrayType
  arrayIndex0
  environment0
  _2))
with
| arraydef_0__dsu =>
    l
| arraydef_0__hasBeenInitialized =>
    [1%Z]
| arraydef_0__result =>
    [0%Z]
end
end)).
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64)
                    8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(λ _0 : arrayIndex0,
   match
     _0 as _1
     return
       (list (arrayType arrayIndex0 environment0 _1))
   with
   | arraydef_0__dsu =>
       <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]> l
   | arraydef_0__hasBeenInitialized => [1%Z]
   | arraydef_0__result => [0%Z]
   end) =
λ _0 : arrayIndex0,
  match decide (_0 = arraydef_0__dsu) with
  | left _1 =>
      eq_rect_r
        (λ _2 : arrayIndex0,
           list
             (arrayType arrayIndex0 environment0 _2))
        (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]> l)
        _1
  | right _ =>
      match
        _0 as _2
        return
          (list
             (arrayType arrayIndex0 environment0 _2))
      with
      | arraydef_0__dsu => l
      | arraydef_0__hasBeenInitialized => [1%Z]
      | arraydef_0__result => [0%Z]
      end
  end
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64)
                    8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
hh: (λ _0 : arrayIndex0,
   match
     _0 as _1
     return
       (list
          (arrayType arrayIndex0 environment0 _1))
   with
   | arraydef_0__dsu =>
       <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
         l
   | arraydef_0__hasBeenInitialized => [1%Z]
   | arraydef_0__result => [0%Z]
   end) =
λ _0 : arrayIndex0,
  match decide (_0 = arraydef_0__dsu) with
  | left _1 =>
      eq_rect_r
        (λ _2 : arrayIndex0,
           list
             (arrayType arrayIndex0 environment0
                _2))
        (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l) _1
  | right _ =>
      match
        _0 as _2
        return
          (list
             (arrayType arrayIndex0 environment0
                _2))
      with
      | arraydef_0__dsu => l
      | arraydef_0__hasBeenInitialized => [1%Z]
      | arraydef_0__result => [0%Z]
      end
  end
(if
  decide
    (Nat.lt (Z.to_nat (100 - Z.of_nat n - 1))
       (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match decide (_0 = arraydef_0__dsu) with
       | left _1 =>
           eq_rect_r
             (λ _2 : arrayIndex0,
                list
                  (arrayType arrayIndex0 environment0
                    _2))
             (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
                l) _1
       | right _ =>
           match
             _0 as _2
             return
               (list
                  (arrayType arrayIndex0 environment0
                    _2))
           with
           | arraydef_0__dsu => l
           | arraydef_0__hasBeenInitialized => [1%Z]
           | arraydef_0__result => [0%Z]
           end
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       (loop n
          (λ _0 : nat,
             dropWithinLoop
               (liftToWithinLoop
                  (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _0 - 1)%Z >>=
                   λ _1 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue ...,
                    Done
                    (WithLocalVariables arrayIndex0
                    ... varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue ...)
                    _3)) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
        λ _ : (), nextCommands))
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) {
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64)
                    8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(λ _0 : arrayIndex0,
   match
     _0 as _1
     return
       (list (arrayType arrayIndex0 environment0 _1))
   with
   | arraydef_0__dsu =>
       <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]> l
   | arraydef_0__hasBeenInitialized => [1%Z]
   | arraydef_0__result => [0%Z]
   end) =
λ _0 : arrayIndex0,
  match decide (_0 = arraydef_0__dsu) with
  | left _1 =>
      eq_rect_r
        (λ _2 : arrayIndex0,
           list
             (arrayType arrayIndex0 environment0 _2))
        (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]> l)
        _1
  | right _ =>
      match
        _0 as _2
        return
          (list
             (arrayType arrayIndex0 environment0 _2))
      with
      | arraydef_0__dsu => l
      | arraydef_0__hasBeenInitialized => [1%Z]
      | arraydef_0__result => [0%Z]
      end
  end apply functional_extensionality_dep.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64)
                    8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
∀ _0 : arrayIndex0,
  match
    _0 as _1
    return
      (list (arrayType arrayIndex0 environment0 _1))
  with
  | arraydef_0__dsu =>
      <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]> l
  | arraydef_0__hasBeenInitialized => [1%Z]
  | arraydef_0__result => [0%Z]
  end =
  match decide (_0 = arraydef_0__dsu) with
  | left _1 =>
      eq_rect_r
        (λ _2 : arrayIndex0,
           list
             (arrayType arrayIndex0 environment0 _2))
        (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]> l)
        _1
  | right _ =>
      match
        _0 as _2
        return
          (list
             (arrayType arrayIndex0 environment0 _2))
      with
      | arraydef_0__dsu => l
      | arraydef_0__hasBeenInitialized => [1%Z]
      | arraydef_0__result => [0%Z]
      end
  end intro x.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64)
                    8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
x: arrayIndex0
match
  x as _0
  return
    (list (arrayType arrayIndex0 environment0 _0))
with
| arraydef_0__dsu =>
    <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]> l
| arraydef_0__hasBeenInitialized => [1%Z]
| arraydef_0__result => [0%Z]
end =
match decide (x = arraydef_0__dsu) with
| left _0 =>
    eq_rect_r
      (λ _1 : arrayIndex0,
         list (arrayType arrayIndex0 environment0 _1))
      (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]> l)
      _0
| right _ =>
    match
      x as _1
      return
        (list (arrayType arrayIndex0 environment0 _1))
    with
    | arraydef_0__dsu => l
    | arraydef_0__hasBeenInitialized => [1%Z]
    | arraydef_0__result => [0%Z]
    end
end destruct x; simpl; easy. }
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64)
                    8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
hh: (λ _0 : arrayIndex0,
   match
     _0 as _1
     return
       (list
          (arrayType arrayIndex0 environment0 _1))
   with
   | arraydef_0__dsu =>
       <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
         l
   | arraydef_0__hasBeenInitialized => [1%Z]
   | arraydef_0__result => [0%Z]
   end) =
λ _0 : arrayIndex0,
  match decide (_0 = arraydef_0__dsu) with
  | left _1 =>
      eq_rect_r
        (λ _2 : arrayIndex0,
           list
             (arrayType arrayIndex0 environment0
                _2))
        (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l) _1
  | right _ =>
      match
        _0 as _2
        return
          (list
             (arrayType arrayIndex0 environment0
                _2))
      with
      | arraydef_0__dsu => l
      | arraydef_0__hasBeenInitialized => [1%Z]
      | arraydef_0__result => [0%Z]
      end
  end
(if
  decide
    (Nat.lt (Z.to_nat (100 - Z.of_nat n - 1))
       (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match decide (_0 = arraydef_0__dsu) with
       | left _1 =>
           eq_rect_r
             (λ _2 : arrayIndex0,
                list
                  (arrayType arrayIndex0 environment0
                    _2))
             (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
                l) _1
       | right _ =>
           match
             _0 as _2
             return
               (list
                  (arrayType arrayIndex0 environment0
                    _2))
           with
           | arraydef_0__dsu => l
           | arraydef_0__hasBeenInitialized => [1%Z]
           | arraydef_0__result => [0%Z]
           end
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       (loop n
          (λ _0 : nat,
             dropWithinLoop
               (liftToWithinLoop
                  (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _0 - 1)%Z >>=
                   λ _1 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue ...,
                    Done
                    (WithLocalVariables arrayIndex0
                    ... varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue ...)
                    _3)) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
        λ _ : (), nextCommands))
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite <- hh.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64)
                    8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
hh: (λ _0 : arrayIndex0,
   match
     _0 as _1
     return
       (list
          (arrayType arrayIndex0 environment0 _1))
   with
   | arraydef_0__dsu =>
       <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
         l
   | arraydef_0__hasBeenInitialized => [1%Z]
   | arraydef_0__result => [0%Z]
   end) =
λ _0 : arrayIndex0,
  match decide (_0 = arraydef_0__dsu) with
  | left _1 =>
      eq_rect_r
        (λ _2 : arrayIndex0,
           list
             (arrayType arrayIndex0 environment0
                _2))
        (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l) _1
  | right _ =>
      match
        _0 as _2
        return
          (list
             (arrayType arrayIndex0 environment0
                _2))
      with
      | arraydef_0__dsu => l
      | arraydef_0__hasBeenInitialized => [1%Z]
      | arraydef_0__result => [0%Z]
      end
  end
(if
  decide
    (Nat.lt (Z.to_nat (100 - Z.of_nat n - 1))
       (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
             l
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       (loop n
          (λ _0 : nat,
             dropWithinLoop
               (liftToWithinLoop
                  (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _0 - 1)%Z >>=
                   λ _1 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue ...,
                    Done
                    (WithLocalVariables arrayIndex0
                    ... varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue ...)
                    _3)) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
        λ _ : (), nextCommands))
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) clear hh.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z
                    (coerceInt
                    (coerceInt (- (1)) 64)
                    8) >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(if
  decide
    (Nat.lt (Z.to_nat (100 - Z.of_nat n - 1))
       (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
             l
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       (loop n
          (λ _0 : nat,
             dropWithinLoop
               (liftToWithinLoop
                  (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _0 - 1)%Z >>=
                   λ _1 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue ...,
                    Done
                    (WithLocalVariables arrayIndex0
                    ... varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue ...)
                    _3)) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
        λ _ : (), nextCommands))
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite !(ltac:(cbv; reflexivity) : (coerceInt (coerceInt (Z.opp 1) 64) 8) = 255%Z) in previous.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(if
  decide
    (Nat.lt (Z.to_nat (100 - Z.of_nat n - 1))
       (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
             l
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       (loop n
          (λ _0 : nat,
             dropWithinLoop
               (liftToWithinLoop
                  (Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    (100 - Z.of_nat _0 - 1)%Z >>=
                   λ _1 : Z,
                    Done
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue Z
                    255%Z >>=
                    λ _2 : Z,
                    Dispatch
                    (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays arrayIndex0 
                    (...) varsfuncdef_0__main 
                    (...)))
                    (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main
                    (Store arrayIndex0 
                    (...) arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue ...,
                    Done
                    (WithLocalVariables arrayIndex0
                    ... varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue ...)
                    _3)) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0) varsfuncdef_0__main)
                    withinLoopReturnValue () ())) >>=
        λ _ : (), nextCommands))
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite previous.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(if
  decide
    (Nat.lt (Z.to_nat (100 - Z.of_nat n - 1))
       (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - n)
             (<[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
                l) ++ 
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite insert_take_drop; [| lia].
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(if
  decide
    (Nat.lt (Z.to_nat (100 - Z.of_nat n - 1))
       (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - n)
             (take (Z.to_nat (100 - Z.of_nat n - 1)) l ++
              255%Z
              :: drop
                   (S
                    (Z.to_nat (100 - Z.of_nat n - 1)))
                   l) ++ 
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite (ltac:(lia) : Z.to_nat (100 - Z.of_nat n - 1) = 100 - S n).
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - n)
             (take (100 - S n) l ++
              255%Z :: drop (S (100 - S n)) l) ++
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite (ltac:(intros; listsEqual) : forall a b c, a ++ b :: c = (a ++ [b]) ++ c).
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - n)
             ((take (100 - S n) l ++ [255%Z]) ++
              drop (S (100 - S n)) l) ++
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) pose proof take_app_length (take (100 - S n) l ++ [255%Z]) (drop (S (100 - S n)) l) as step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
step: take (length (take (100 - S n) l ++ [255%Z]))
  ((take (100 - S n) l ++ [255%Z]) ++
   drop (S (100 - S n)) l) =
take (100 - S n) l ++ [255%Z]
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - n)
             ((take (100 - S n) l ++ [255%Z]) ++
              drop (S (100 - S n)) l) ++
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite app_length in step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
step: take
  (length (take (100 - S n) l) + length [255%Z])
  ((take (100 - S n) l ++ [255%Z]) ++
   drop (S (100 - S n)) l) =
take (100 - S n) l ++ [255%Z]
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - n)
             ((take (100 - S n) l ++ [255%Z]) ++
              drop (S (100 - S n)) l) ++
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite (ltac:(easy) : length [255%Z] = 1) in step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
step: take (length (take (100 - S n) l) + 1)
  ((take (100 - S n) l ++ [255%Z]) ++
   drop (S (100 - S n)) l) =
take (100 - S n) l ++ [255%Z]
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - n)
             ((take (100 - S n) l ++ [255%Z]) ++
              drop (S (100 - S n)) l) ++
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite take_length in step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
step: take ((100 - S n) `min` length l + 1)
  ((take (100 - S n) l ++ [255%Z]) ++
   drop (S (100 - S n)) l) =
take (100 - S n) l ++ [255%Z]
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - n)
             ((take (100 - S n) l ++ [255%Z]) ++
              drop (S (100 - S n)) l) ++
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite (ltac:(lia) : (100 - S n) `min` length l = 100 - S n) in step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
step: take (100 - S n + 1)
  ((take (100 - S n) l ++ [255%Z]) ++
   drop (S (100 - S n)) l) =
take (100 - S n) l ++ [255%Z]
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - n)
             ((take (100 - S n) l ++ [255%Z]) ++
              drop (S (100 - S n)) l) ++
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite (ltac:(lia) : 100 - S n + 1 = 100 - n) in step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
step: take (100 - n)
  ((take (100 - S n) l ++ [255%Z]) ++
   drop (S (100 - S n)) l) =
take (100 - S n) l ++ [255%Z]
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - n)
             ((take (100 - S n) l ++ [255%Z]) ++
              drop (S (100 - S n)) l) ++
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
step: take (100 - n)
  ((take (100 - S n) l ++ [255%Z]) ++
   drop (S (100 - S n)) l) =
take (100 - S n) l ++ [255%Z]
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           (take (100 - S n) l ++ [255%Z]) ++
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) clear step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           (take (100 - S n) l ++ [255%Z]) ++
           repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite (ltac:(intros; listsEqual) : forall a b c, (a ++ [b]) ++ c = a ++ (b :: c)).
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - S n) l ++
           255%Z :: repeat 255%Z n
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite (ltac:(easy) : _ :: repeat _ _ = repeat 255%Z (S n)).
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
(if decide (Nat.lt (100 - S n) (length l))
 then
  invokeContractAux (repeat 1%Z 20) 
    (repeat 0%Z 20) 0 state state [
    a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
    (arrayType arrayIndex0 environment0)
    (λ _0 : arrayIndex0,
       match
         _0 as _1
         return
           (list
              (arrayType arrayIndex0 environment0 _1))
       with
       | arraydef_0__dsu =>
           take (100 - S n) l ++ repeat 255%Z (S n)
       | arraydef_0__hasBeenInitialized => [1%Z]
       | arraydef_0__result => [0%Z]
       end)
    (funcdef_0__main
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
    (eliminateLocalVariables
       (λ _ : varsfuncdef_0__main, false)
       (λ _ : varsfuncdef_0__main, 0%Z)
       (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
       nextCommands)
 else Some ([], state)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) case_decide as hIf; [reflexivity |].
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
hIf: ¬ Nat.lt (100 - S n) (length l)
Some ([], state) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) pose proof (ltac:(lia) : @length (arrayType arrayIndex0 environment0 arraydef_0__dsu) l <= 100 - S n) as step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
hIf: ¬ Nat.lt (100 - S n) (length l)
step: length l ≤ 100 - S n
Some ([], state) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) simpl in step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
hIf: ¬ Nat.lt (100 - S n) (length l)
step: length l ≤ 99 - n
Some ([], state) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) rewrite hL in step.
a, b: Z
n: nat
nextCommands: Action
  (WithLocalVariables arrayIndex0
     (arrayType arrayIndex0
        environment0)
     varsfuncdef_0__main)
  withLocalVariablesReturnValue ()
h: S n ≤ 100
l: list Z
hL: length l = 100
previous: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         <[Z.to_nat (100 - Z.of_nat n - 1):=255%Z]>
           l
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20)
     (loop n
        (λ _0 : nat,
           dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                   withLocalVariablesReturnValue
                   Z
                   (100 - Z.of_nat _0 - 1)%Z >>=
                 λ _1 : Z,
                   Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    Z 255%Z >>=
                   λ _2 : Z,
                    Dispatch
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2)))
                    (DoWithArrays
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main
                    (Store arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    arraydef_0__dsu _1 _2))
                    (λ _3 : 
                    withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...)),
                    Done
                    (WithLocalVariables
                    arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                    withLocalVariablesReturnValue
                    (withLocalVariablesReturnValue
                    (DoWithArrays
                    arrayIndex0 
                    (...)
                    varsfuncdef_0__main
                    (...))) _3)) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                    environment0)
                    varsfuncdef_0__main)
                  withinLoopReturnValue ()
                  ())) >>=
      λ _ : (), nextCommands)) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - n)
           (<[Z.to_nat
                (100 - Z.of_nat n - 1):=255%Z]>
              l) ++ repeat 255%Z n
     | arraydef_0__hasBeenInitialized =>
         [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main,
        repeat 0%Z 20) nextCommands)
hIf: ¬ Nat.lt (100 - S n) (length l)
step: 100 ≤ 99 - n
Some ([], state) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state state [
  a; b] 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu =>
         take (100 - S n) l ++ repeat 255%Z (S n)
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [0%Z]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20)
     nextCommands) lia.
Qed.

Lemma runAncestor1 (dsu : list Slot) state1 state2 (hL : length dsu = 100) (hM : Z.to_nat (dsuLeafCount dsu) = length dsu) (h1 : noIllegalIndices dsu) (h2 : withoutCyclesN dsu (length dsu)) (whatever2 a : Z) (hLe1 : Z.le 0 a) (hLt1 : Z.lt a 100) (communication : list Z) continuation continuation2 whatever n (hN : n <= 100) : invokeContractAux (repeat 1%Z 20) (repeat 0%Z 20) 0 state1 state2
  communication 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0) (λ _0 : arrayIndex0,
  match
  _0 as _1
return
  (list
  (arrayType
  arrayIndex0
  environment0
  _1))
with
| arraydef_0__dsu =>
    convertToArray dsu
| arraydef_0__hasBeenInitialized =>
    [1%Z]
| arraydef_0__result =>
    [whatever]
end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
  (λ _ : varsfuncdef_0__main, 0%Z)
  (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
  (λ _ : varsfuncdef_0__ancestor, false) (λ _0 : varsfuncdef_0__ancestor,
  match _0
with
| vardef_0__ancestor_vertex =>
    whatever2
| vardef_0__ancestor_work =>
    a
end)
  (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
  (loop n
  (λ _ : nat,
  dropWithinLoop
  (liftToWithinLoop
  (numberLocalGet arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor
  vardef_0__ancestor_work >>=
λ _0 : withLocalVariablesReturnValue
  (NumberLocalGet arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor
  vardef_0__ancestor_work),
  Done
  (WithLocalVariables arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor)
  withLocalVariablesReturnValue Z
  (coerceInt _0 64) >>=
λ _1 : Z,
  retrieve arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor arraydef_0__dsu
  _1 >>=
λ _2 : arrayType arrayIndex0 environment0
  arraydef_0__dsu,
  Done
  (WithLocalVariables arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor)
  withLocalVariablesReturnValue Z
  (coerceInt 0 8) >>=
λ _3 : Z,
  Done
  (WithLocalVariables arrayIndex0
  (arrayType arrayIndex0
  environment0)
  varsfuncdef_0__ancestor)
  withLocalVariablesReturnValue bool
  (bool_decide
  (toSigned _2 8 < toSigned _3 8)%Z)) >>=
λ _0 : bool,
  (if _0
then
 break arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor >>=
λ _ : (),
  Done
  (WithinLoop arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor)
  withinLoopReturnValue ()
  ()
else
 Done
  (WithinLoop arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor)
  withinLoopReturnValue ()
  ()) >>=
λ _ : (),
  liftToWithinLoop
  (numberLocalGet arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor
  vardef_0__ancestor_work >>=
λ _1 : withLocalVariablesReturnValue
  (NumberLocalGet arrayIndex0
  (arrayType arrayIndex0
  environment0)
  varsfuncdef_0__ancestor
  vardef_0__ancestor_work),
  Done
  (WithLocalVariables arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor)
  withLocalVariablesReturnValue Z
  (coerceInt _1 64) >>=
λ _2 : Z,
  retrieve arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor
  arraydef_0__dsu _2 >>=
λ _3 : arrayType arrayIndex0
  environment0 arraydef_0__dsu,
  numberLocalSet arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor
  vardef_0__ancestor_work
  _3) >>=
λ _ : (),
  Done
  (WithinLoop arrayIndex0
  (arrayType arrayIndex0 environment0)
  varsfuncdef_0__ancestor)
  withinLoopReturnValue ()
  ())) >>= continuation) >>= continuation2) = invokeContractAux (repeat 1%Z 20) (repeat 0%Z 20) 0 state1 state2
  communication 1 arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0) (λ _0 : arrayIndex0,
  match
  _0 as _1
return
  (list
  (arrayType
  arrayIndex0
  environment0
  _1))
with
| arraydef_0__dsu =>
    convertToArray dsu
| arraydef_0__hasBeenInitialized =>
    [1%Z]
| arraydef_0__result =>
    [whatever]
end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
  (λ _ : varsfuncdef_0__main, 0%Z)
  (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
  (λ _ : varsfuncdef_0__ancestor, false) (λ _0 : varsfuncdef_0__ancestor,
  match _0
with
| vardef_0__ancestor_vertex =>
    whatever2
| vardef_0__ancestor_work =>
    Z.of_nat (ancestor dsu n (Z.to_nat a))
end)
  (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20) (continuation tt) >>= continuation2).
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2, a: Z
hLe1: (0 ≤ a)%Z
hLt1: (a < 100)%Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: n ≤ 100
invokeContractAux (repeat 1%Z 20) (repeat 0%Z 20) 0
  state1 state2 communication 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work >>=
                 λ _0 : withLocalVariablesReturnValue
                          (NumberLocalGet arrayIndex0
                             (arrayType arrayIndex0
                                environment0)
                             varsfuncdef_0__ancestor
                             vardef_0__ancestor_work),
                   Done
                     (WithLocalVariables arrayIndex0
                        (arrayType arrayIndex0
                           environment0)
                        varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                          environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : arrayType arrayIndex0
                              environment0
                              arraydef_0__dsu,
                       Done
                         (WithLocalVariables
                            arrayIndex0
                            (arrayType arrayIndex0
                               environment0)
                            varsfuncdef_0__ancestor)
                         withLocalVariablesReturnValue
                         Z (coerceInt 0 8) >>=
                       λ _3 : Z,
                         Done
                           (WithLocalVariables
                              arrayIndex0 (...)
                              varsfuncdef_0__ancestor)
                           withLocalVariablesReturnValue
                           bool (bool_decide (...)%Z)) >>=
              λ _0 : bool,
                (if _0
                 then
                  break arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                         (arrayType arrayIndex0
                            environment0)
                         varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()
                 else
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                          environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()) >>=
                λ _ : (),
                  liftToWithinLoop
                    (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                          environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                     λ _1 : withLocalVariablesReturnValue
                              (NumberLocalGet
                                 arrayIndex0 (...)
                                 varsfuncdef_0__ancestor
                                 vardef_0__ancestor_work),
                       Done
                         (WithLocalVariables
                            arrayIndex0
                            (arrayType arrayIndex0
                               environment0)
                            varsfuncdef_0__ancestor)
                         withLocalVariablesReturnValue
                         Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                         retrieve arrayIndex0
                           (arrayType arrayIndex0
                              environment0)
                           varsfuncdef_0__ancestor
                           arraydef_0__dsu _2 >>=
                         λ _3 : ...,
                           numberLocalSet arrayIndex0
                             (...)
                             varsfuncdef_0__ancestor
                             vardef_0__ancestor_work
                             _3) >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                         (arrayType arrayIndex0
                            environment0)
                         varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) (repeat 0%Z 20) 0
  state1 state2 communication 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu n (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
Proof.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2, a: Z
hLe1: (0 ≤ a)%Z
hLt1: (a < 100)%Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: n ≤ 100
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work >>=
                 λ _0 : 
                    withLocalVariablesReturnValue
                      (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                   Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
              λ _0 : bool,
                (if _0
                 then
                  break arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()
                 else
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()) >>=
                λ _ : (),
                  liftToWithinLoop
                    (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                     λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       numberLocalSet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu n (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  revert a hLt1 hLe1.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: n ≤ 100
∀ _0 : Z,
  (_0 < 100)%Z
  → (0 ≤ _0)%Z
    → invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state1 state2 communication
        1 arrayIndex0 arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0 environment0
                     _2))
           with
           | arraydef_0__dsu => convertToArray dsu
           | arraydef_0__hasBeenInitialized => [1%Z]
           | arraydef_0__result => [whatever]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__ancestor, false)
           (λ _1 : varsfuncdef_0__ancestor,
              match _1 with
              | vardef_0__ancestor_vertex => whatever2
              | vardef_0__ancestor_work => _0
              end)
           (λ _ : varsfuncdef_0__ancestor,
              repeat 0%Z 20)
           (loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done 
                       (...)
                       withLocalVariablesReturnValue Z
                       (...) >>=
                       λ ...,
                       Done ...
                       withLocalVariablesReturnValue
                       bool ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 
                       (...) varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       ...,
                       Done 
                       (...)
                       withLocalVariablesReturnValue Z
                       (...) >>= 
                       λ ..., ... >>= ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
            continuation) >>= continuation2) =
      invokeContractAux 
        (repeat 1%Z 20) 
        (repeat 0%Z 20) 0 state1 state2 communication
        1 arrayIndex0 arrayIndexEqualityDecidable0
        (arrayType arrayIndex0 environment0)
        (λ _1 : arrayIndex0,
           match
             _1 as _2
             return
               (list
                  (arrayType arrayIndex0 environment0
                     _2))
           with
           | arraydef_0__dsu => convertToArray dsu
           | arraydef_0__hasBeenInitialized => [1%Z]
           | arraydef_0__result => [whatever]
           end)
        (funcdef_0__main
           (λ _ : varsfuncdef_0__main, false)
           (λ _ : varsfuncdef_0__main, 0%Z)
           (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
        (eliminateLocalVariables
           (λ _ : varsfuncdef_0__ancestor, false)
           (λ _1 : varsfuncdef_0__ancestor,
              match _1 with
              | vardef_0__ancestor_vertex => whatever2
              | vardef_0__ancestor_work =>
                  Z.of_nat
                    (ancestor dsu n (Z.to_nat _0))
              end)
           (λ _ : varsfuncdef_0__ancestor,
              repeat 0%Z 20) 
           (continuation ()) >>= continuation2) induction n as [| n IH]; intros a hLt1 hLe1.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
hN: 0 ≤ 100
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop 0
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work >>=
                 λ _0 : 
                    withLocalVariablesReturnValue
                      (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                   Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
              λ _0 : bool,
                (if _0
                 then
                  break arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()
                 else
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()) >>=
                λ _ : (),
                  liftToWithinLoop
                    (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                     λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       numberLocalSet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu 0 (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop (S n)
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work >>=
                 λ _0 : 
                    withLocalVariablesReturnValue
                      (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                   Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
              λ _0 : bool,
                (if _0
                 then
                  break arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()
                 else
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()) >>=
                λ _ : (),
                  liftToWithinLoop
                    (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                     λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       numberLocalSet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  -
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
hN: 0 ≤ 100
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop 0
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work >>=
                 λ _0 : 
                    withLocalVariablesReturnValue
                      (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                   Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
              λ _0 : bool,
                (if _0
                 then
                  break arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()
                 else
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()) >>=
                λ _ : (),
                  liftToWithinLoop
                    (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                     λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       numberLocalSet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu 0 (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite (ltac:(simpl; reflexivity) : loop 0 _ = _), (ltac:(simpl; reflexivity) : ancestor dsu 0 _ = _), Z2Nat.id, leftIdentity; [reflexivity | lia].
  -
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop (S n)
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work >>=
                 λ _0 : 
                    withLocalVariablesReturnValue
                      (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                   Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
              λ _0 : bool,
                (if _0
                 then
                  break arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()
                 else
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()) >>=
                λ _ : (),
                  liftToWithinLoop
                    (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                     λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       numberLocalSet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite (ltac:(simpl; reflexivity) : loop (S _) _ = _).
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (numberLocalGet arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor
              vardef_0__ancestor_work >>=
            λ _0 : withLocalVariablesReturnValue
                     (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
              Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withLocalVariablesReturnValue Z
                (coerceInt _0 64) >>=
              λ _1 : Z,
                retrieve arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  arraydef_0__dsu _1 >>=
                λ _2 : arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt 0 8) >>=
                  λ _3 : Z,
                    Done
                      (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withLocalVariablesReturnValue
                      bool
                      (bool_decide
                       (toSigned _2 8 < toSigned _3 8)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      (λ _0 : LoopOutcome,
         match _0 with
         | KeepGoing =>
             loop n
               (λ _ : nat,
                  dropWithinLoop
                    (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0 
                       (...) varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ ..., ... >>= ...) >>=
                     λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done 
                       (...) withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0 
                       (...) varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ ..., ... >>= ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 
                       (...) varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
         | Stop =>
             Done
               (WithLocalVariables arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withLocalVariablesReturnValue () ()
         end) >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) unfold numberLocalGet at 1.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Dispatch
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue
              (withLocalVariablesReturnValue
                 (NumberLocalGet arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor
                    vardef_0__ancestor_work))
              (NumberLocalGet arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor
                 vardef_0__ancestor_work)
              (λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                 Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue
                   (withLocalVariablesReturnValue
                      (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work)) _0) >>=
            λ _0 : withLocalVariablesReturnValue
                     (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
              Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withLocalVariablesReturnValue Z
                (coerceInt _0 64) >>=
              λ _1 : Z,
                retrieve arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  arraydef_0__dsu _1 >>=
                λ _2 : arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt 0 8) >>=
                  λ _3 : Z,
                    Done
                      (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withLocalVariablesReturnValue
                      bool
                      (bool_decide
                       (toSigned _2 8 < toSigned _3 8)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      (λ _0 : LoopOutcome,
         match _0 with
         | KeepGoing =>
             loop n
               (λ _ : nat,
                  dropWithinLoop
                    (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0 
                       (...) varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ ..., ... >>= ...) >>=
                     λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done 
                       (...) withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0 
                       (...) varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ ..., ... >>= ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 
                       (...) varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
         | Stop =>
             Done
               (WithLocalVariables arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withLocalVariablesReturnValue () ()
         end) >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite <- !bindAssoc, liftToWithinLoopBind, <- !bindAssoc, dropWithinLoopLiftToWithinLoop, <- !bindAssoc.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (Dispatch
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor)
        withLocalVariablesReturnValue
        (withLocalVariablesReturnValue
           (NumberLocalGet arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor
              vardef_0__ancestor_work))
        (NumberLocalGet arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor
           vardef_0__ancestor_work)
        (λ _0 : withLocalVariablesReturnValue
                  (NumberLocalGet arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     vardef_0__ancestor_work),
           Done
             (WithLocalVariables arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withLocalVariablesReturnValue
             (withLocalVariablesReturnValue
                (NumberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work)) _0) >>=
      λ _0 : withLocalVariablesReturnValue
               (NumberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work),
        dropWithinLoop
          (liftToWithinLoop
             (Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withLocalVariablesReturnValue Z
                (coerceInt _0 64) >>=
              λ _1 : Z,
                retrieve arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  arraydef_0__dsu _1 >>=
                λ _2 : arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt 0 8) >>=
                  λ _3 : Z,
                    Done
                      (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withLocalVariablesReturnValue
                      bool
                      (bool_decide
                       (toSigned _2 8 < toSigned _3 8)%Z)) >>=
           λ _1 : bool,
             (if _1
              then
               break arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor >>=
               λ _ : (),
                 Done
                   (WithinLoop arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withinLoopReturnValue () ()
              else
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
             λ _ : (),
               liftToWithinLoop
                 (numberLocalGet arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor
                    vardef_0__ancestor_work >>=
                  λ _2 : 
                     withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                    Done
                      (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withLocalVariablesReturnValue Z
                      (coerceInt _2 64) >>=
                    λ _3 : Z,
                      retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>=
                      λ _4 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _4) >>=
               λ _ : (),
                 Done
                   (WithinLoop arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withinLoopReturnValue () ()) >>=
        λ _1 : LoopOutcome,
          match _1 with
          | KeepGoing =>
              loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       (...),
                       Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 
                       (...) varsfuncdef_0__ancestor >>=
                       λ ...,
                       Done ... withinLoopReturnValue
                       () ()
                       else
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
          | Stop =>
              Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withLocalVariablesReturnValue () ()
          end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) pose proof @pushNumberGet2 arrayIndex0 (arrayType arrayIndex0 environment0) varsfuncdef_0__ancestor _ (λ _ : varsfuncdef_0__ancestor, false) (λ _1 : varsfuncdef_0__ancestor,
  match
  _1
with
| vardef_0__ancestor_vertex =>
    whatever2
| vardef_0__ancestor_work =>
    a
end) (λ _ : varsfuncdef_0__ancestor,
  repeat
  0%Z 20) as step.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: ∀ (_0 : varsfuncdef_0__ancestor) 
  (_1 : withLocalVariablesReturnValue
          (NumberLocalGet arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__ancestor _0)
        → Action
            (WithLocalVariables arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__ancestor)
            withLocalVariablesReturnValue ()),
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__ancestor, false)
    (λ _2 : varsfuncdef_0__ancestor,
       match _2 with
       | vardef_0__ancestor_vertex => whatever2
       | vardef_0__ancestor_work => a
       end)
    (λ _ : varsfuncdef_0__ancestor,
       repeat 0%Z 20)
    (Dispatch
       (WithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor)
       withLocalVariablesReturnValue
       (withLocalVariablesReturnValue
          (NumberLocalGet arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__ancestor _0))
       (NumberLocalGet arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor _0)
       (λ _2 : withLocalVariablesReturnValue
                 (NumberLocalGet arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor _0),
          Done
            (WithLocalVariables arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__ancestor)
            withLocalVariablesReturnValue
            (withLocalVariablesReturnValue
               (NumberLocalGet arrayIndex0
                  (arrayType arrayIndex0
                     environment0)
                  varsfuncdef_0__ancestor _0))
            _2) >>= _1) =
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__ancestor, false)
    (λ _2 : varsfuncdef_0__ancestor,
       match _2 with
       | vardef_0__ancestor_vertex => whatever2
       | vardef_0__ancestor_work => a
       end)
    (λ _ : varsfuncdef_0__ancestor,
       repeat 0%Z 20)
    (_1
       ((λ _2 : varsfuncdef_0__ancestor,
           match _2 with
           | vardef_0__ancestor_vertex =>
               whatever2
           | vardef_0__ancestor_work => a
           end) _0))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (Dispatch
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor)
        withLocalVariablesReturnValue
        (withLocalVariablesReturnValue
           (NumberLocalGet arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor
              vardef_0__ancestor_work))
        (NumberLocalGet arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor
           vardef_0__ancestor_work)
        (λ _0 : withLocalVariablesReturnValue
                  (NumberLocalGet arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     vardef_0__ancestor_work),
           Done
             (WithLocalVariables arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withLocalVariablesReturnValue
             (withLocalVariablesReturnValue
                (NumberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work)) _0) >>=
      λ _0 : withLocalVariablesReturnValue
               (NumberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work),
        dropWithinLoop
          (liftToWithinLoop
             (Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withLocalVariablesReturnValue Z
                (coerceInt _0 64) >>=
              λ _1 : Z,
                retrieve arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  arraydef_0__dsu _1 >>=
                λ _2 : arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt 0 8) >>=
                  λ _3 : Z,
                    Done
                      (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withLocalVariablesReturnValue
                      bool
                      (bool_decide
                       (toSigned _2 8 < toSigned _3 8)%Z)) >>=
           λ _1 : bool,
             (if _1
              then
               break arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor >>=
               λ _ : (),
                 Done
                   (WithinLoop arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withinLoopReturnValue () ()
              else
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
             λ _ : (),
               liftToWithinLoop
                 (numberLocalGet arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor
                    vardef_0__ancestor_work >>=
                  λ _2 : 
                     withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                    Done
                      (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withLocalVariablesReturnValue Z
                      (coerceInt _2 64) >>=
                    λ _3 : Z,
                      retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>=
                      λ _4 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _4) >>=
               λ _ : (),
                 Done
                   (WithinLoop arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withinLoopReturnValue () ()) >>=
        λ _1 : LoopOutcome,
          match _1 with
          | KeepGoing =>
              loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       (...),
                       Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 
                       (...) varsfuncdef_0__ancestor >>=
                       λ ...,
                       Done ... withinLoopReturnValue
                       () ()
                       else
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
          | Stop =>
              Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withLocalVariablesReturnValue () ()
          end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite step.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: ∀ (_0 : varsfuncdef_0__ancestor) 
  (_1 : withLocalVariablesReturnValue
          (NumberLocalGet arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__ancestor _0)
        → Action
            (WithLocalVariables arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__ancestor)
            withLocalVariablesReturnValue ()),
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__ancestor, false)
    (λ _2 : varsfuncdef_0__ancestor,
       match _2 with
       | vardef_0__ancestor_vertex => whatever2
       | vardef_0__ancestor_work => a
       end)
    (λ _ : varsfuncdef_0__ancestor,
       repeat 0%Z 20)
    (Dispatch
       (WithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor)
       withLocalVariablesReturnValue
       (withLocalVariablesReturnValue
          (NumberLocalGet arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__ancestor _0))
       (NumberLocalGet arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor _0)
       (λ _2 : withLocalVariablesReturnValue
                 (NumberLocalGet arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor _0),
          Done
            (WithLocalVariables arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__ancestor)
            withLocalVariablesReturnValue
            (withLocalVariablesReturnValue
               (NumberLocalGet arrayIndex0
                  (arrayType arrayIndex0
                     environment0)
                  varsfuncdef_0__ancestor _0))
            _2) >>= _1) =
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__ancestor, false)
    (λ _2 : varsfuncdef_0__ancestor,
       match _2 with
       | vardef_0__ancestor_vertex => whatever2
       | vardef_0__ancestor_work => a
       end)
    (λ _ : varsfuncdef_0__ancestor,
       repeat 0%Z 20)
    (_1
       ((λ _2 : varsfuncdef_0__ancestor,
           match _2 with
           | vardef_0__ancestor_vertex =>
               whatever2
           | vardef_0__ancestor_work => a
           end) _0))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue Z
              (coerceInt a 64) >>=
            λ _0 : Z,
              retrieve arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor
                arraydef_0__dsu _0 >>=
              λ _1 : arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                Done
                  (WithLocalVariables arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withLocalVariablesReturnValue Z
                  (coerceInt 0 8) >>=
                λ _2 : Z,
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue bool
                    (bool_decide
                       (toSigned _1 8 < toSigned _2 8)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) clear step.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue Z
              (coerceInt a 64) >>=
            λ _0 : Z,
              retrieve arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor
                arraydef_0__dsu _0 >>=
              λ _1 : arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                Done
                  (WithLocalVariables arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withLocalVariablesReturnValue Z
                  (coerceInt 0 8) >>=
                λ _2 : Z,
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue bool
                    (bool_decide
                       (toSigned _1 8 < toSigned _2 8)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  assert (step : coerceInt a 64 = a).
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
coerceInt a 64 = a
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue Z
              (coerceInt a 64) >>=
            λ _0 : Z,
              retrieve arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor
                arraydef_0__dsu _0 >>=
              λ _1 : arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                Done
                  (WithLocalVariables arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withLocalVariablesReturnValue Z
                  (coerceInt 0 8) >>=
                λ _2 : Z,
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue bool
                    (bool_decide
                       (toSigned _1 8 < toSigned _2 8)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  {
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
coerceInt a 64 = a revert hLe1 hLt1.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
(0 ≤ a)%Z → (a < 100)%Z → coerceInt a 64 = a clear.
a: Z
(0 ≤ a)%Z → (a < 100)%Z → coerceInt a 64 = a intros h1 h2.
a: Z
h1: (0 ≤ a)%Z
h2: (a < 100)%Z
coerceInt a 64 = a unfold coerceInt.
a: Z
h1: (0 ≤ a)%Z
h2: (a < 100)%Z
(a `mod` 2 ^ 64)%Z = a rewrite Z.mod_small.
a: Z
h1: (0 ≤ a)%Z
h2: (a < 100)%Z
a = a
a: Z
h1: (0 ≤ a)%Z
h2: (a < 100)%Z
(0 ≤ a < 2 ^ 64)%Z {
a: Z
h1: (0 ≤ a)%Z
h2: (a < 100)%Z
a = a reflexivity. }
a: Z
h1: (0 ≤ a)%Z
h2: (a < 100)%Z
(0 ≤ a < 2 ^ 64)%Z lia. }
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue Z
              (coerceInt a 64) >>=
            λ _0 : Z,
              retrieve arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor
                arraydef_0__dsu _0 >>=
              λ _1 : arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                Done
                  (WithLocalVariables arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withLocalVariablesReturnValue Z
                  (coerceInt 0 8) >>=
                λ _2 : Z,
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue bool
                    (bool_decide
                       (toSigned _1 8 < toSigned _2 8)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  rewrite step, !leftIdentity, liftToWithinLoopBind, <- !bindAssoc, dropWithinLoopLiftToWithinLoop.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (retrieve arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__ancestor arraydef_0__dsu a >>=
      (λ _0 : withLocalVariablesReturnValue
                (DoWithArrays arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   (Retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0) arraydef_0__dsu a)),
         dropWithinLoop
           (liftToWithinLoop
              (Done
                 (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withLocalVariablesReturnValue Z
                 (coerceInt 0 8) >>=
               λ _1 : Z,
                 Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue bool
                   (bool_decide
                      (toSigned _0 8 < toSigned _1 8)%Z)) >>=
            λ _1 : bool,
              (if _1
               then
                break arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()
               else
                Done
                  (WithinLoop arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withinLoopReturnValue () ()) >>=
              λ _ : (),
                liftToWithinLoop
                  (numberLocalGet arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     vardef_0__ancestor_work >>=
                   λ _2 : 
                      withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                     Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                     λ _3 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>=
                       λ _4 : ...,
                       numberLocalSet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _4) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ())) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) unfold retrieve at 1.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (Dispatch
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor)
        withLocalVariablesReturnValue
        (withLocalVariablesReturnValue
           (DoWithArrays arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor
              (Retrieve arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 arraydef_0__dsu a)))
        (DoWithArrays arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor
           (Retrieve arrayIndex0
              (arrayType arrayIndex0 environment0)
              arraydef_0__dsu a))
        (λ _0 : withLocalVariablesReturnValue
                  (DoWithArrays arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     (Retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0) arraydef_0__dsu a)),
           Done
             (WithLocalVariables arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withLocalVariablesReturnValue
             (withLocalVariablesReturnValue
                (DoWithArrays arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   (Retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0) arraydef_0__dsu a)))
             _0) >>=
      (λ _0 : withLocalVariablesReturnValue
                (DoWithArrays arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   (Retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0) arraydef_0__dsu a)),
         dropWithinLoop
           (liftToWithinLoop
              (Done
                 (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withLocalVariablesReturnValue Z
                 (coerceInt 0 8) >>=
               λ _1 : Z,
                 Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue bool
                   (bool_decide
                      (toSigned _0 8 < toSigned _1 8)%Z)) >>=
            λ _1 : bool,
              (if _1
               then
                break arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()
               else
                Done
                  (WithinLoop arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withinLoopReturnValue () ()) >>=
              λ _ : (),
                liftToWithinLoop
                  (numberLocalGet arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     vardef_0__ancestor_work >>=
                   λ _2 : 
                      withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                     Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                     λ _3 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>=
                       λ _4 : ...,
                       numberLocalSet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _4) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ())) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  pose proof pushDispatch2 (λ _ : varsfuncdef_0__ancestor, false)
  (λ _0 : varsfuncdef_0__ancestor,
     match _0 with
     | vardef_0__ancestor_vertex => whatever2
     | vardef_0__ancestor_work => a
     end) (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20) (Retrieve arrayIndex0 (arrayType arrayIndex0 environment0)
     arraydef_0__dsu a) as step2.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
step2: ∀ _0 : withLocalVariablesReturnValue
         (DoWithArrays arrayIndex0
            (arrayType arrayIndex0
               environment0)
            varsfuncdef_0__ancestor
            (Retrieve arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               arraydef_0__dsu a))
       → Action
           (WithLocalVariables arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__ancestor)
           withLocalVariablesReturnValue (),
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__ancestor, false)
    (λ _1 : varsfuncdef_0__ancestor,
       match _1 with
       | vardef_0__ancestor_vertex =>
           whatever2
       | vardef_0__ancestor_work => a
       end)
    (λ _ : varsfuncdef_0__ancestor,
       repeat 0%Z 20)
    (Dispatch
       (WithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor)
       withLocalVariablesReturnValue
       (withLocalVariablesReturnValue
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__ancestor
             (Retrieve arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu a)))
       (DoWithArrays arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor
          (Retrieve arrayIndex0
             (arrayType arrayIndex0
                environment0) arraydef_0__dsu
             a))
       (λ _1 : withLocalVariablesReturnValue
                 (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor
                    (Retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       arraydef_0__dsu a)),
          Done
            (WithLocalVariables arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__ancestor)
            withLocalVariablesReturnValue
            (withLocalVariablesReturnValue
               (DoWithArrays arrayIndex0
                  (arrayType arrayIndex0
                     environment0)
                  varsfuncdef_0__ancestor
                  (Retrieve arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     arraydef_0__dsu a))) _1) >>=
     _0) =
  Dispatch
    (WithArrays arrayIndex0
       (arrayType arrayIndex0 environment0))
    withArraysReturnValue ()
    (Retrieve arrayIndex0
       (arrayType arrayIndex0 environment0)
       arraydef_0__dsu a)
    (λ _1 : withArraysReturnValue
              (Retrieve arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 arraydef_0__dsu a),
       eliminateLocalVariables
         (λ _ : varsfuncdef_0__ancestor, false)
         (λ _2 : varsfuncdef_0__ancestor,
            match _2 with
            | vardef_0__ancestor_vertex =>
                whatever2
            | vardef_0__ancestor_work => a
            end)
         (λ _ : varsfuncdef_0__ancestor,
            repeat 0%Z 20) 
         (_0 _1))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (Dispatch
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor)
        withLocalVariablesReturnValue
        (withLocalVariablesReturnValue
           (DoWithArrays arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor
              (Retrieve arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 arraydef_0__dsu a)))
        (DoWithArrays arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor
           (Retrieve arrayIndex0
              (arrayType arrayIndex0 environment0)
              arraydef_0__dsu a))
        (λ _0 : withLocalVariablesReturnValue
                  (DoWithArrays arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     (Retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0) arraydef_0__dsu a)),
           Done
             (WithLocalVariables arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withLocalVariablesReturnValue
             (withLocalVariablesReturnValue
                (DoWithArrays arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   (Retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0) arraydef_0__dsu a)))
             _0) >>=
      (λ _0 : withLocalVariablesReturnValue
                (DoWithArrays arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   (Retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0) arraydef_0__dsu a)),
         dropWithinLoop
           (liftToWithinLoop
              (Done
                 (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withLocalVariablesReturnValue Z
                 (coerceInt 0 8) >>=
               λ _1 : Z,
                 Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue bool
                   (bool_decide
                      (toSigned _0 8 < toSigned _1 8)%Z)) >>=
            λ _1 : bool,
              (if _1
               then
                break arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()
               else
                Done
                  (WithinLoop arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withinLoopReturnValue () ()) >>=
              λ _ : (),
                liftToWithinLoop
                  (numberLocalGet arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     vardef_0__ancestor_work >>=
                   λ _2 : 
                      withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                     Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                     λ _3 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>=
                       λ _4 : ...,
                       numberLocalSet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _4) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ())) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite <- !bindAssoc, step2.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
step2: ∀ _0 : withLocalVariablesReturnValue
         (DoWithArrays arrayIndex0
            (arrayType arrayIndex0
               environment0)
            varsfuncdef_0__ancestor
            (Retrieve arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               arraydef_0__dsu a))
       → Action
           (WithLocalVariables arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__ancestor)
           withLocalVariablesReturnValue (),
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__ancestor, false)
    (λ _1 : varsfuncdef_0__ancestor,
       match _1 with
       | vardef_0__ancestor_vertex =>
           whatever2
       | vardef_0__ancestor_work => a
       end)
    (λ _ : varsfuncdef_0__ancestor,
       repeat 0%Z 20)
    (Dispatch
       (WithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor)
       withLocalVariablesReturnValue
       (withLocalVariablesReturnValue
          (DoWithArrays arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__ancestor
             (Retrieve arrayIndex0
                (arrayType arrayIndex0
                   environment0)
                arraydef_0__dsu a)))
       (DoWithArrays arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor
          (Retrieve arrayIndex0
             (arrayType arrayIndex0
                environment0) arraydef_0__dsu
             a))
       (λ _1 : withLocalVariablesReturnValue
                 (DoWithArrays arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor
                    (Retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       arraydef_0__dsu a)),
          Done
            (WithLocalVariables arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__ancestor)
            withLocalVariablesReturnValue
            (withLocalVariablesReturnValue
               (DoWithArrays arrayIndex0
                  (arrayType arrayIndex0
                     environment0)
                  varsfuncdef_0__ancestor
                  (Retrieve arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     arraydef_0__dsu a))) _1) >>=
     _0) =
  Dispatch
    (WithArrays arrayIndex0
       (arrayType arrayIndex0 environment0))
    withArraysReturnValue ()
    (Retrieve arrayIndex0
       (arrayType arrayIndex0 environment0)
       arraydef_0__dsu a)
    (λ _1 : withArraysReturnValue
              (Retrieve arrayIndex0
                 (arrayType arrayIndex0
                    environment0)
                 arraydef_0__dsu a),
       eliminateLocalVariables
         (λ _ : varsfuncdef_0__ancestor, false)
         (λ _2 : varsfuncdef_0__ancestor,
            match _2 with
            | vardef_0__ancestor_vertex =>
                whatever2
            | vardef_0__ancestor_work => a
            end)
         (λ _ : varsfuncdef_0__ancestor,
            repeat 0%Z 20) 
         (_0 _1))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (Dispatch
     (WithArrays arrayIndex0
        (arrayType arrayIndex0 environment0))
     withArraysReturnValue ()
     (Retrieve arrayIndex0
        (arrayType arrayIndex0 environment0)
        arraydef_0__dsu a)
     (λ _0 : withArraysReturnValue
               (Retrieve arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  arraydef_0__dsu a),
        eliminateLocalVariables
          (λ _ : varsfuncdef_0__ancestor, false)
          (λ _1 : varsfuncdef_0__ancestor,
             match _1 with
             | vardef_0__ancestor_vertex => whatever2
             | vardef_0__ancestor_work => a
             end)
          (λ _ : varsfuncdef_0__ancestor,
             repeat 0%Z 20)
          (dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue Z
                   (coerceInt 0 8) >>=
                 λ _1 : Z,
                   Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue
                     bool
                     (bool_decide
                       (toSigned _0 8 < toSigned _1 8)%Z)) >>=
              λ _1 : bool,
                (if _1
                 then
                  break arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()
                 else
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()) >>=
                λ _ : (),
                  liftToWithinLoop
                    (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                     λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>=
                       λ _4 : ...,
                       numberLocalSet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _4) >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
           λ _1 : LoopOutcome,
             match _1 with
             | KeepGoing =>
                 loop n
                   (λ _ : nat,
                      dropWithinLoop
                       (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       ...,
                       Done 
                       (...)
                       withLocalVariablesReturnValue Z
                       (...) >>= 
                       λ ..., ... >>= ...) >>=
                       λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 ...
                       varsfuncdef_0__ancestor >>=
                       ..., ...
                       else
                       Done 
                       (...) withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop (... >>= ...) >>=
                       λ _ : (),
                       Done 
                       (...) withinLoopReturnValue ()
                       ()))
             | Stop =>
                 Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue () ()
             end >>= continuation)) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) clear step2.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (Dispatch
     (WithArrays arrayIndex0
        (arrayType arrayIndex0 environment0))
     withArraysReturnValue ()
     (Retrieve arrayIndex0
        (arrayType arrayIndex0 environment0)
        arraydef_0__dsu a)
     (λ _0 : withArraysReturnValue
               (Retrieve arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  arraydef_0__dsu a),
        eliminateLocalVariables
          (λ _ : varsfuncdef_0__ancestor, false)
          (λ _1 : varsfuncdef_0__ancestor,
             match _1 with
             | vardef_0__ancestor_vertex => whatever2
             | vardef_0__ancestor_work => a
             end)
          (λ _ : varsfuncdef_0__ancestor,
             repeat 0%Z 20)
          (dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue Z
                   (coerceInt 0 8) >>=
                 λ _1 : Z,
                   Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue
                     bool
                     (bool_decide
                       (toSigned _0 8 < toSigned _1 8)%Z)) >>=
              λ _1 : bool,
                (if _1
                 then
                  break arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()
                 else
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()) >>=
                λ _ : (),
                  liftToWithinLoop
                    (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                     λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>=
                       λ _4 : ...,
                       numberLocalSet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _4) >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
           λ _1 : LoopOutcome,
             match _1 with
             | KeepGoing =>
                 loop n
                   (λ _ : nat,
                      dropWithinLoop
                       (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       ...,
                       Done 
                       (...)
                       withLocalVariablesReturnValue Z
                       (...) >>= 
                       λ ..., ... >>= ...) >>=
                       λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 ...
                       varsfuncdef_0__ancestor >>=
                       ..., ...
                       else
                       Done 
                       (...) withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop (... >>= ...) >>=
                       λ _ : (),
                       Done 
                       (...) withinLoopReturnValue ()
                       ()))
             | Stop =>
                 Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue () ()
             end >>= continuation)) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  rewrite (ltac:(simpl; reflexivity) : forall effect continuation f, bind (Dispatch _ _ _ effect continuation) f = _).
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (Dispatch
     (WithArrays arrayIndex0
        (arrayType arrayIndex0 environment0))
     withArraysReturnValue ()
     (Retrieve arrayIndex0
        (arrayType arrayIndex0 environment0)
        arraydef_0__dsu a)
     (λ _0 : withArraysReturnValue
               (Retrieve arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  arraydef_0__dsu a),
        eliminateLocalVariables
          (λ _ : varsfuncdef_0__ancestor, false)
          (λ _1 : varsfuncdef_0__ancestor,
             match _1 with
             | vardef_0__ancestor_vertex => whatever2
             | vardef_0__ancestor_work => a
             end)
          (λ _ : varsfuncdef_0__ancestor,
             repeat 0%Z 20)
          (dropWithinLoop
             (liftToWithinLoop
                (Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue Z
                   (coerceInt 0 8) >>=
                 λ _1 : Z,
                   Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue
                     bool
                     (bool_decide
                       (toSigned _0 8 < toSigned _1 8)%Z)) >>=
              λ _1 : bool,
                (if _1
                 then
                  break arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()
                 else
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ()) >>=
                λ _ : (),
                  liftToWithinLoop
                    (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                     λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>=
                       λ _4 : ...,
                       numberLocalSet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _4) >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
           λ _1 : LoopOutcome,
             match _1 with
             | KeepGoing =>
                 loop n
                   (λ _ : nat,
                      dropWithinLoop
                       (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       ...,
                       Done 
                       (...)
                       withLocalVariablesReturnValue Z
                       (...) >>= 
                       λ ..., ... >>= ...) >>=
                       λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 ...
                       varsfuncdef_0__ancestor >>=
                       ..., ...
                       else
                       Done 
                       (...) withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop (... >>= ...) >>=
                       λ _ : (),
                       Done 
                       (...) withinLoopReturnValue ()
                       ()))
             | Stop =>
                 Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue () ()
             end >>= continuation) >>= continuation2)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) autorewrite with advance_program.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
match
  decide
    (Nat.lt (Z.to_nat a) (length (convertToArray dsu)))
with
| left _0 =>
    invokeContractAux (repeat 1%Z 20) 
      (repeat 0%Z 20) 0 state1 state2 communication 1
      arrayIndex0 arrayIndexEqualityDecidable0
      (arrayType arrayIndex0 environment0)
      (λ _1 : arrayIndex0,
         match
           _1 as _2
           return
             (list
                (arrayType arrayIndex0 environment0 _2))
         with
         | arraydef_0__dsu => convertToArray dsu
         | arraydef_0__hasBeenInitialized => [1%Z]
         | arraydef_0__result => [whatever]
         end)
      (funcdef_0__main
         (λ _ : varsfuncdef_0__main, false)
         (λ _ : varsfuncdef_0__main, 0%Z)
         (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
      (eliminateLocalVariables
         (λ _ : varsfuncdef_0__ancestor, false)
         (λ _1 : varsfuncdef_0__ancestor,
            match _1 with
            | vardef_0__ancestor_vertex => whatever2
            | vardef_0__ancestor_work => a
            end)
         (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
         (dropWithinLoop
            (liftToWithinLoop
               (Done
                  (WithLocalVariables arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withLocalVariablesReturnValue Z
                  (coerceInt 0 8) >>=
                λ _1 : Z,
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue bool
                    (bool_decide
                       (toSigned
                       (nth_lt 
                       (convertToArray dsu)
                       (Z.to_nat a) _0) 8 <
                       toSigned _1 8)%Z)) >>=
             λ _1 : bool,
               (if _1
                then
                 break arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor >>=
                 λ _ : (),
                   Done
                     (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withinLoopReturnValue () ()
                else
                 Done
                   (WithinLoop arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withinLoopReturnValue () ()) >>=
               λ _ : (),
                 liftToWithinLoop
                   (numberLocalGet arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      vardef_0__ancestor_work >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                      λ _3 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>=
                       λ _4 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _4) >>=
                 λ _ : (),
                   Done
                     (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withinLoopReturnValue () ()) >>=
          λ _1 : LoopOutcome,
            match _1 with
            | KeepGoing =>
                loop n
                  (λ _ : nat,
                     dropWithinLoop
                       (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       (...),
                       Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                       λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 
                       (...) varsfuncdef_0__ancestor >>=
                       λ ...,
                       Done ... withinLoopReturnValue
                       () ()
                       else
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
            | Stop =>
                Done
                  (WithLocalVariables arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withLocalVariablesReturnValue () ()
            end >>= continuation) >>= continuation2)
| right _ => Some ([], state1)
end =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  case_decide as hs; [| rewrite lengthConvert in hs; lia].
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue Z
              (coerceInt 0 8) >>=
            λ _0 : Z,
              Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withLocalVariablesReturnValue bool
                (bool_decide
                   (toSigned
                      (nth_lt 
                       (convertToArray dsu)
                       (Z.to_nat a) hs) 8 <
                    toSigned _0 8)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  rewrite !leftIdentity, (ltac:(easy) : toSigned (coerceInt 0%Z 8) 8 = 0%Z).
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool
              (bool_decide
                 (toSigned
                    (nth_lt 
                       (convertToArray dsu)
                       (Z.to_nat a) hs) 8 < 0)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  rewrite (ltac:(easy) : @nth_lt (arrayType arrayIndex0 environment0 arraydef_0__dsu) (convertToArray dsu) (Z.to_nat a) hs = @nth_lt Z (convertToArray dsu) (Z.to_nat a) hs), (nth_lt_default (convertToArray dsu) (Z.to_nat a) hs 0%Z), nthConvert; [| lia].
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool
              (bool_decide
                 (toSigned
                    match
                      nth 
                       (Z.to_nat a) dsu
                       (Ancestor Unit)
                    with
                    | ReferTo _0 => Z.of_nat _0
                    | Ancestor _0 =>
                       256 - Z.of_nat (leafCount _0)
                    end 8 < 0)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  remember (nth (Z.to_nat a) dsu (Ancestor Unit)) as g eqn:hg.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
g: Slot
hg: g = nth (Z.to_nat a) dsu (Ancestor Unit)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool
              (bool_decide
                 (toSigned
                    match g with
                    | ReferTo _0 => Z.of_nat _0
                    | Ancestor _0 =>
                       256 - Z.of_nat (leafCount _0)
                    end 8 < 0)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) symmetry in hg.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
g: Slot
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool
              (bool_decide
                 (toSigned
                    match g with
                    | ReferTo _0 => Z.of_nat _0
                    | Ancestor _0 =>
                       256 - Z.of_nat (leafCount _0)
                    end 8 < 0)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  destruct g as [g | g]; clear hs.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool
              (bool_decide
                 (toSigned (Z.of_nat g) 8 < 0)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool
              (bool_decide
                 (toSigned
                    (256 - Z.of_nat (leafCount g)) 8 <
                  0)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
  +
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool
              (bool_decide
                 (toSigned (Z.of_nat g) 8 < 0)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) case_bool_decide as hs.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: (toSigned (Z.of_nat g) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool true) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ¬ (toSigned (Z.of_nat g) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) {
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: (toSigned (Z.of_nat g) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool true) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) unfold toSigned in hs.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ((if decide (Z.of_nat g < 2 ^ (8 - 1))
  then Z.of_nat g
  else Z.of_nat g - 2 ^ 8) < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool true) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) pose proof h1 (Z.to_nat a) g hg.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ((if decide (Z.of_nat g < 2 ^ (8 - 1))
  then Z.of_nat g
  else Z.of_nat g - 2 ^ 8) < 0)%Z
_0: g < length dsu
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool true) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) case_decide; [| lia].
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
_1: (Z.of_nat g < 2 ^ (8 - 1))%Z
hs: (Z.of_nat g < 0)%Z
_0: g < length dsu
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool true) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) lia. }
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ¬ (toSigned (Z.of_nat g) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite (ltac:(easy) : liftToWithinLoop
  (Done
     (WithLocalVariables arrayIndex0 (arrayType arrayIndex0 environment0)
        varsfuncdef_0__ancestor) withLocalVariablesReturnValue bool false) = Done _ _ bool false), !leftIdentity.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ¬ (toSigned (Z.of_nat g) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (numberLocalGet arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor
              vardef_0__ancestor_work >>=
            λ _0 : withLocalVariablesReturnValue
                     (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
              Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withLocalVariablesReturnValue Z
                (coerceInt _0 64) >>=
              λ _1 : Z,
                retrieve arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  arraydef_0__dsu _1 >>=
                λ _2 : arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                  numberLocalSet arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor
                    vardef_0__ancestor_work _2) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
    rewrite liftToWithinLoopBind, <- !bindAssoc, dropWithinLoopLiftToWithinLoop.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ¬ (toSigned (Z.of_nat g) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (numberLocalGet arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__ancestor
        vardef_0__ancestor_work >>=
      (λ _0 : withLocalVariablesReturnValue
                (NumberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work),
         dropWithinLoop
           (liftToWithinLoop
              (Done
                 (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withLocalVariablesReturnValue Z
                 (coerceInt _0 64) >>=
               λ _1 : Z,
                 retrieve arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   arraydef_0__dsu _1 >>=
                 λ _2 : 
                    arrayType arrayIndex0 environment0
                      arraydef_0__dsu,
                   numberLocalSet arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     vardef_0__ancestor_work _2) >>=
            λ _ : (),
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withinLoopReturnValue () ())) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       liftToWithinLoop
                       (Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) unfold numberLocalGet at 1.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ¬ (toSigned (Z.of_nat g) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (Dispatch
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor)
        withLocalVariablesReturnValue
        (withLocalVariablesReturnValue
           (NumberLocalGet arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor
              vardef_0__ancestor_work))
        (NumberLocalGet arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor
           vardef_0__ancestor_work)
        (λ _0 : withLocalVariablesReturnValue
                  (NumberLocalGet arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     vardef_0__ancestor_work),
           Done
             (WithLocalVariables arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withLocalVariablesReturnValue
             (withLocalVariablesReturnValue
                (NumberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work)) _0) >>=
      (λ _0 : withLocalVariablesReturnValue
                (NumberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work),
         dropWithinLoop
           (liftToWithinLoop
              (Done
                 (WithLocalVariables arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withLocalVariablesReturnValue Z
                 (coerceInt _0 64) >>=
               λ _1 : Z,
                 retrieve arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   arraydef_0__dsu _1 >>=
                 λ _2 : 
                    arrayType arrayIndex0 environment0
                      arraydef_0__dsu,
                   numberLocalSet arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     vardef_0__ancestor_work _2) >>=
            λ _ : (),
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withinLoopReturnValue () ())) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       liftToWithinLoop
                       (Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite <- !bindAssoc, pushNumberGet2, !step, !leftIdentity, liftToWithinLoopBind, <- !bindAssoc, dropWithinLoopLiftToWithinLoop.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ¬ (toSigned (Z.of_nat g) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (retrieve arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__ancestor arraydef_0__dsu a >>=
      (λ _0 : withLocalVariablesReturnValue
                (DoWithArrays arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   (Retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0) arraydef_0__dsu a)),
         dropWithinLoop
           (liftToWithinLoop
              (numberLocalSet arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor
                 vardef_0__ancestor_work _0) >>=
            λ _ : (),
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withinLoopReturnValue () ())) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       liftToWithinLoop
                       (Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) unfold retrieve at 1.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ¬ (toSigned (Z.of_nat g) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (Dispatch
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor)
        withLocalVariablesReturnValue
        (withLocalVariablesReturnValue
           (DoWithArrays arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor
              (Retrieve arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 arraydef_0__dsu a)))
        (DoWithArrays arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor
           (Retrieve arrayIndex0
              (arrayType arrayIndex0 environment0)
              arraydef_0__dsu a))
        (λ _0 : withLocalVariablesReturnValue
                  (DoWithArrays arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     (Retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0) arraydef_0__dsu a)),
           Done
             (WithLocalVariables arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withLocalVariablesReturnValue
             (withLocalVariablesReturnValue
                (DoWithArrays arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   (Retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0) arraydef_0__dsu a)))
             _0) >>=
      (λ _0 : withLocalVariablesReturnValue
                (DoWithArrays arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   (Retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0) arraydef_0__dsu a)),
         dropWithinLoop
           (liftToWithinLoop
              (numberLocalSet arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor
                 vardef_0__ancestor_work _0) >>=
            λ _ : (),
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withinLoopReturnValue () ())) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       liftToWithinLoop
                       (Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite <- !bindAssoc, pushDispatch2.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ¬ (toSigned (Z.of_nat g) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (Dispatch
     (WithArrays arrayIndex0
        (arrayType arrayIndex0 environment0))
     withArraysReturnValue ()
     (Retrieve arrayIndex0
        (arrayType arrayIndex0 environment0)
        arraydef_0__dsu a)
     (λ _0 : withArraysReturnValue
               (Retrieve arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  arraydef_0__dsu a),
        eliminateLocalVariables
          (λ _ : varsfuncdef_0__ancestor, false)
          (λ _1 : varsfuncdef_0__ancestor,
             match _1 with
             | vardef_0__ancestor_vertex => whatever2
             | vardef_0__ancestor_work => a
             end)
          (λ _ : varsfuncdef_0__ancestor,
             repeat 0%Z 20)
          (dropWithinLoop
             (liftToWithinLoop
                (numberLocalSet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work _0) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withinLoopReturnValue () ()) >>=
           λ _1 : LoopOutcome,
             match _1 with
             | KeepGoing =>
                 loop n
                   (λ _ : nat,
                      dropWithinLoop
                       (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       ...,
                       Done 
                       (...)
                       withLocalVariablesReturnValue Z
                       (...) >>= 
                       λ ..., ... >>= ...) >>=
                       λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 ...
                       varsfuncdef_0__ancestor >>=
                       ..., ...
                       else
                       Done 
                       (...) withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _3 : ...,
                       liftToWithinLoop ... >>=
                       ..., ...))
             | Stop =>
                 Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue () ()
             end >>= continuation)) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite (ltac:(simpl; reflexivity) : forall effect continuation f, bind (Dispatch _ _ _ effect continuation) f = _).
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ¬ (toSigned (Z.of_nat g) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (Dispatch
     (WithArrays arrayIndex0
        (arrayType arrayIndex0 environment0))
     withArraysReturnValue ()
     (Retrieve arrayIndex0
        (arrayType arrayIndex0 environment0)
        arraydef_0__dsu a)
     (λ _0 : withArraysReturnValue
               (Retrieve arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  arraydef_0__dsu a),
        eliminateLocalVariables
          (λ _ : varsfuncdef_0__ancestor, false)
          (λ _1 : varsfuncdef_0__ancestor,
             match _1 with
             | vardef_0__ancestor_vertex => whatever2
             | vardef_0__ancestor_work => a
             end)
          (λ _ : varsfuncdef_0__ancestor,
             repeat 0%Z 20)
          (dropWithinLoop
             (liftToWithinLoop
                (numberLocalSet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work _0) >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withinLoopReturnValue () ()) >>=
           λ _1 : LoopOutcome,
             match _1 with
             | KeepGoing =>
                 loop n
                   (λ _ : nat,
                      dropWithinLoop
                       (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       ...,
                       Done 
                       (...)
                       withLocalVariablesReturnValue Z
                       (...) >>= 
                       λ ..., ... >>= ...) >>=
                       λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 ...
                       varsfuncdef_0__ancestor >>=
                       ..., ...
                       else
                       Done 
                       (...) withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _3 : ...,
                       liftToWithinLoop ... >>=
                       ..., ...))
             | Stop =>
                 Done
                   (WithLocalVariables arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor)
                   withLocalVariablesReturnValue () ()
             end >>= continuation) >>= continuation2)) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) autorewrite with advance_program.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: ¬ (toSigned (Z.of_nat g) 8 < 0)%Z
match
  decide
    (Nat.lt (Z.to_nat a) (length (convertToArray dsu)))
with
| left _0 =>
    invokeContractAux (repeat 1%Z 20) 
      (repeat 0%Z 20) 0 state1 state2 communication 1
      arrayIndex0 arrayIndexEqualityDecidable0
      (arrayType arrayIndex0 environment0)
      (λ _1 : arrayIndex0,
         match
           _1 as _2
           return
             (list
                (arrayType arrayIndex0 environment0 _2))
         with
         | arraydef_0__dsu => convertToArray dsu
         | arraydef_0__hasBeenInitialized => [1%Z]
         | arraydef_0__result => [whatever]
         end)
      (funcdef_0__main
         (λ _ : varsfuncdef_0__main, false)
         (λ _ : varsfuncdef_0__main, 0%Z)
         (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
      (eliminateLocalVariables
         (λ _ : varsfuncdef_0__ancestor, false)
         (λ _1 : varsfuncdef_0__ancestor,
            match _1 with
            | vardef_0__ancestor_vertex => whatever2
            | vardef_0__ancestor_work => a
            end)
         (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
         (dropWithinLoop
            (liftToWithinLoop
               (numberLocalSet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work
                  (nth_lt 
                     (convertToArray dsu) 
                     (Z.to_nat a) _0)) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
          λ _1 : LoopOutcome,
            match _1 with
            | KeepGoing =>
                loop n
                  (λ _ : nat,
                     dropWithinLoop
                       (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       (...),
                       Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                       λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 
                       (...) varsfuncdef_0__ancestor >>=
                       λ ...,
                       Done ... withinLoopReturnValue
                       () ()
                       else
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _3 : 
                       withLocalVariablesReturnValue
                       ...,
                       liftToWithinLoop (...) >>=
                       λ ...,
                       Done ... withinLoopReturnValue
                       () ()))
            | Stop =>
                Done
                  (WithLocalVariables arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withLocalVariablesReturnValue () ()
            end >>= continuation) >>= continuation2)
| right _ => Some ([], state1)
end =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) clear hs.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
match
  decide
    (Nat.lt (Z.to_nat a) (length (convertToArray dsu)))
with
| left _0 =>
    invokeContractAux (repeat 1%Z 20) 
      (repeat 0%Z 20) 0 state1 state2 communication 1
      arrayIndex0 arrayIndexEqualityDecidable0
      (arrayType arrayIndex0 environment0)
      (λ _1 : arrayIndex0,
         match
           _1 as _2
           return
             (list
                (arrayType arrayIndex0 environment0 _2))
         with
         | arraydef_0__dsu => convertToArray dsu
         | arraydef_0__hasBeenInitialized => [1%Z]
         | arraydef_0__result => [whatever]
         end)
      (funcdef_0__main
         (λ _ : varsfuncdef_0__main, false)
         (λ _ : varsfuncdef_0__main, 0%Z)
         (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
      (eliminateLocalVariables
         (λ _ : varsfuncdef_0__ancestor, false)
         (λ _1 : varsfuncdef_0__ancestor,
            match _1 with
            | vardef_0__ancestor_vertex => whatever2
            | vardef_0__ancestor_work => a
            end)
         (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
         (dropWithinLoop
            (liftToWithinLoop
               (numberLocalSet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work
                  (nth_lt 
                     (convertToArray dsu) 
                     (Z.to_nat a) _0)) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
          λ _1 : LoopOutcome,
            match _1 with
            | KeepGoing =>
                loop n
                  (λ _ : nat,
                     dropWithinLoop
                       (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       (...),
                       Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                       λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 
                       (...) varsfuncdef_0__ancestor >>=
                       λ ...,
                       Done ... withinLoopReturnValue
                       () ()
                       else
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _3 : 
                       withLocalVariablesReturnValue
                       ...,
                       liftToWithinLoop (...) >>=
                       λ ...,
                       Done ... withinLoopReturnValue
                       () ()))
            | Stop =>
                Done
                  (WithLocalVariables arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withLocalVariablesReturnValue () ()
            end >>= continuation) >>= continuation2)
| right _ => Some ([], state1)
end =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) case_decide as hs; [| rewrite lengthConvert in hs; lia].
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (numberLocalSet arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor
              vardef_0__ancestor_work
              (nth_lt (convertToArray dsu)
                 (Z.to_nat a) hs)) >>=
         λ _ : (),
           Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       liftToWithinLoop
                       (Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite liftToWithinLoopBind, <- !bindAssoc, dropWithinLoopLiftToWithinLoop, <- !bindAssoc.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (numberLocalSet arrayIndex0
        (arrayType arrayIndex0 environment0)
        varsfuncdef_0__ancestor
        vardef_0__ancestor_work
        (nth_lt (convertToArray dsu) (Z.to_nat a) hs) >>=
      λ _ : (),
        dropWithinLoop
          (Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withinLoopReturnValue () ()) >>=
        λ _0 : LoopOutcome,
          match _0 with
          | KeepGoing =>
              loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                      λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>= 
                       ..., ...) >>=
                       λ _2 : bool,
                       (if _2
                       then ... >>= ...
                       else
                       Done ... withinLoopReturnValue
                       () ()) >>=
                       λ _ : (),
                       liftToWithinLoop (...) >>=
                       λ ..., ... >>= ...))
          | Stop =>
              Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withLocalVariablesReturnValue () ()
          end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) unfold numberLocalSet at 1.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (Dispatch
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor)
        withLocalVariablesReturnValue
        (withLocalVariablesReturnValue
           (NumberLocalSet arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor
              vardef_0__ancestor_work
              (nth_lt (convertToArray dsu)
                 (Z.to_nat a) hs)))
        (NumberLocalSet arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor
           vardef_0__ancestor_work
           (nth_lt (convertToArray dsu) 
              (Z.to_nat a) hs))
        (λ _0 : withLocalVariablesReturnValue
                  (NumberLocalSet arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     vardef_0__ancestor_work
                     (nth_lt 
                       (convertToArray dsu)
                       (Z.to_nat a) hs)),
           Done
             (WithLocalVariables arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withLocalVariablesReturnValue
             (withLocalVariablesReturnValue
                (NumberLocalSet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work
                   (nth_lt 
                      (convertToArray dsu)
                      (Z.to_nat a) hs))) _0) >>=
      λ _ : (),
        dropWithinLoop
          (Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withinLoopReturnValue () ()) >>=
        λ _0 : LoopOutcome,
          match _0 with
          | KeepGoing =>
              loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                      λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>= 
                       ..., ...) >>=
                       λ _2 : bool,
                       (if _2
                       then ... >>= ...
                       else
                       Done ... withinLoopReturnValue
                       () ()) >>=
                       λ _ : (),
                       liftToWithinLoop (...) >>=
                       λ ..., ... >>= ...))
          | Stop =>
              Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withLocalVariablesReturnValue () ()
          end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) pose proof (@pushNumberSet2 arrayIndex0 (arrayType arrayIndex0 environment0) varsfuncdef_0__ancestor _ (λ _ : varsfuncdef_0__ancestor, false)
    (λ _0 : varsfuncdef_0__ancestor,
       match _0 with
       | vardef_0__ancestor_vertex => whatever2
       | vardef_0__ancestor_work => a
       end) (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20) vardef_0__ancestor_work) (nth_lt (convertToArray dsu) (Z.to_nat a) hs) as step3.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: ∀ _0 : withLocalVariablesReturnValue
         (NumberLocalSet arrayIndex0
            (arrayType arrayIndex0
               environment0)
            varsfuncdef_0__ancestor
            vardef_0__ancestor_work
            (nth_lt (convertToArray dsu)
               (Z.to_nat a) hs))
       → Action
           (WithLocalVariables arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__ancestor)
           withLocalVariablesReturnValue (),
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__ancestor, false)
    (λ _1 : varsfuncdef_0__ancestor,
       match _1 with
       | vardef_0__ancestor_vertex =>
           whatever2
       | vardef_0__ancestor_work => a
       end)
    (λ _ : varsfuncdef_0__ancestor,
       repeat 0%Z 20)
    (Dispatch
       (WithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor)
       withLocalVariablesReturnValue
       (withLocalVariablesReturnValue
          (NumberLocalSet arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__ancestor
             vardef_0__ancestor_work
             (nth_lt (convertToArray dsu)
                (Z.to_nat a) hs)))
       (NumberLocalSet arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor
          vardef_0__ancestor_work
          (nth_lt (convertToArray dsu)
             (Z.to_nat a) hs))
       (λ _1 : withLocalVariablesReturnValue
                 (NumberLocalSet arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor
                    vardef_0__ancestor_work
                    (nth_lt
                       (convertToArray dsu)
                       (Z.to_nat a) hs)),
          Done
            (WithLocalVariables arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__ancestor)
            withLocalVariablesReturnValue
            (withLocalVariablesReturnValue
               (NumberLocalSet arrayIndex0
                  (arrayType arrayIndex0
                     environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work
                  (nth_lt 
                     (convertToArray dsu)
                     (Z.to_nat a) hs))) _1) >>=
     _0) =
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__ancestor, false)
    (update
       (λ _1 : varsfuncdef_0__ancestor,
          match _1 with
          | vardef_0__ancestor_vertex =>
              whatever2
          | vardef_0__ancestor_work => a
          end) vardef_0__ancestor_work
       (nth_lt (convertToArray dsu)
          (Z.to_nat a) hs))
    (λ _ : varsfuncdef_0__ancestor,
       repeat 0%Z 20) (_0 ())
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (Dispatch
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor)
        withLocalVariablesReturnValue
        (withLocalVariablesReturnValue
           (NumberLocalSet arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor
              vardef_0__ancestor_work
              (nth_lt (convertToArray dsu)
                 (Z.to_nat a) hs)))
        (NumberLocalSet arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor
           vardef_0__ancestor_work
           (nth_lt (convertToArray dsu) 
              (Z.to_nat a) hs))
        (λ _0 : withLocalVariablesReturnValue
                  (NumberLocalSet arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor
                     vardef_0__ancestor_work
                     (nth_lt 
                       (convertToArray dsu)
                       (Z.to_nat a) hs)),
           Done
             (WithLocalVariables arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withLocalVariablesReturnValue
             (withLocalVariablesReturnValue
                (NumberLocalSet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work
                   (nth_lt 
                      (convertToArray dsu)
                      (Z.to_nat a) hs))) _0) >>=
      λ _ : (),
        dropWithinLoop
          (Done
             (WithinLoop arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor)
             withinLoopReturnValue () ()) >>=
        λ _0 : LoopOutcome,
          match _0 with
          | KeepGoing =>
              loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                      λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>= 
                       ..., ...) >>=
                       λ _2 : bool,
                       (if _2
                       then ... >>= ...
                       else
                       Done ... withinLoopReturnValue
                       () ()) >>=
                       λ _ : (),
                       liftToWithinLoop (...) >>=
                       λ ..., ... >>= ...))
          | Stop =>
              Done
                (WithLocalVariables arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withLocalVariablesReturnValue () ()
          end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite step3.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: ∀ _0 : withLocalVariablesReturnValue
         (NumberLocalSet arrayIndex0
            (arrayType arrayIndex0
               environment0)
            varsfuncdef_0__ancestor
            vardef_0__ancestor_work
            (nth_lt (convertToArray dsu)
               (Z.to_nat a) hs))
       → Action
           (WithLocalVariables arrayIndex0
              (arrayType arrayIndex0
                 environment0)
              varsfuncdef_0__ancestor)
           withLocalVariablesReturnValue (),
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__ancestor, false)
    (λ _1 : varsfuncdef_0__ancestor,
       match _1 with
       | vardef_0__ancestor_vertex =>
           whatever2
       | vardef_0__ancestor_work => a
       end)
    (λ _ : varsfuncdef_0__ancestor,
       repeat 0%Z 20)
    (Dispatch
       (WithLocalVariables arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor)
       withLocalVariablesReturnValue
       (withLocalVariablesReturnValue
          (NumberLocalSet arrayIndex0
             (arrayType arrayIndex0
                environment0)
             varsfuncdef_0__ancestor
             vardef_0__ancestor_work
             (nth_lt (convertToArray dsu)
                (Z.to_nat a) hs)))
       (NumberLocalSet arrayIndex0
          (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor
          vardef_0__ancestor_work
          (nth_lt (convertToArray dsu)
             (Z.to_nat a) hs))
       (λ _1 : withLocalVariablesReturnValue
                 (NumberLocalSet arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor
                    vardef_0__ancestor_work
                    (nth_lt
                       (convertToArray dsu)
                       (Z.to_nat a) hs)),
          Done
            (WithLocalVariables arrayIndex0
               (arrayType arrayIndex0
                  environment0)
               varsfuncdef_0__ancestor)
            withLocalVariablesReturnValue
            (withLocalVariablesReturnValue
               (NumberLocalSet arrayIndex0
                  (arrayType arrayIndex0
                     environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work
                  (nth_lt 
                     (convertToArray dsu)
                     (Z.to_nat a) hs))) _1) >>=
     _0) =
  eliminateLocalVariables
    (λ _ : varsfuncdef_0__ancestor, false)
    (update
       (λ _1 : varsfuncdef_0__ancestor,
          match _1 with
          | vardef_0__ancestor_vertex =>
              whatever2
          | vardef_0__ancestor_work => a
          end) vardef_0__ancestor_work
       (nth_lt (convertToArray dsu)
          (Z.to_nat a) hs))
    (λ _ : varsfuncdef_0__ancestor,
       repeat 0%Z 20) (_0 ())
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (update
        (λ _0 : varsfuncdef_0__ancestor,
           match _0 with
           | vardef_0__ancestor_vertex => whatever2
           | vardef_0__ancestor_work => a
           end) vardef_0__ancestor_work
        (nth_lt (convertToArray dsu) (Z.to_nat a) hs))
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (Done
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor)
           withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                      λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 
                       (...) varsfuncdef_0__ancestor >>=
                       λ ...,
                       Done ... withinLoopReturnValue
                       () ()
                       else
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _3 : 
                       withLocalVariablesReturnValue
                       ...,
                       liftToWithinLoop (...) >>=
                       λ ...,
                       Done ... withinLoopReturnValue
                       () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) clear step3.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (update
        (λ _0 : varsfuncdef_0__ancestor,
           match _0 with
           | vardef_0__ancestor_vertex => whatever2
           | vardef_0__ancestor_work => a
           end) vardef_0__ancestor_work
        (nth_lt (convertToArray dsu) (Z.to_nat a) hs))
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (Done
           (WithinLoop arrayIndex0
              (arrayType arrayIndex0 environment0)
              varsfuncdef_0__ancestor)
           withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                      λ _2 : bool,
                       (if _2
                       then
                       break arrayIndex0 
                       (...) varsfuncdef_0__ancestor >>=
                       λ ...,
                       Done ... withinLoopReturnValue
                       () ()
                       else
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _3 : 
                       withLocalVariablesReturnValue
                       ...,
                       liftToWithinLoop (...) >>=
                       λ ...,
                       Done ... withinLoopReturnValue
                       () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
    rewrite (ltac:(cbv; reflexivity) : dropWithinLoop
    (Done
       (WithinLoop arrayIndex0 (arrayType arrayIndex0 environment0)
          varsfuncdef_0__ancestor) withinLoopReturnValue () ())  = _), !leftIdentity.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (update
        (λ _0 : varsfuncdef_0__ancestor,
           match _0 with
           | vardef_0__ancestor_vertex => whatever2
           | vardef_0__ancestor_work => a
           end) vardef_0__ancestor_work
        (nth_lt (convertToArray dsu) (Z.to_nat a) hs))
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
    assert (step3 : nth_lt (convertToArray dsu) (Z.to_nat a) hs = Z.of_nat g).
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
nth_lt (convertToArray dsu) (Z.to_nat a) hs =
Z.of_nat g
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: nth_lt (convertToArray dsu) (Z.to_nat a) hs =
Z.of_nat g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (update
        (λ _0 : varsfuncdef_0__ancestor,
           match _0 with
           | vardef_0__ancestor_vertex => whatever2
           | vardef_0__ancestor_work => a
           end) vardef_0__ancestor_work
        (nth_lt (convertToArray dsu) (Z.to_nat a) hs))
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
    {
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
nth_lt (convertToArray dsu) (Z.to_nat a) hs =
Z.of_nat g rewrite (nth_lt_default _ _ _ 0%Z), nthConvert; [| lia].
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
match nth (Z.to_nat a) dsu (Ancestor Unit) with
| ReferTo _0 => Z.of_nat _0
| Ancestor _0 => (256 - Z.of_nat (leafCount _0))%Z
end = Z.of_nat g rewrite hg.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
Z.of_nat g = Z.of_nat g reflexivity. }
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: nth_lt (convertToArray dsu) (Z.to_nat a) hs =
Z.of_nat g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (update
        (λ _0 : varsfuncdef_0__ancestor,
           match _0 with
           | vardef_0__ancestor_vertex => whatever2
           | vardef_0__ancestor_work => a
           end) vardef_0__ancestor_work
        (nth_lt (convertToArray dsu) (Z.to_nat a) hs))
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite step3.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: nth_lt (convertToArray dsu) (Z.to_nat a) hs =
Z.of_nat g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (update
        (λ _0 : varsfuncdef_0__ancestor,
           match _0 with
           | vardef_0__ancestor_vertex => whatever2
           | vardef_0__ancestor_work => a
           end) vardef_0__ancestor_work 
        (Z.of_nat g))
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) clear step3.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (update
        (λ _0 : varsfuncdef_0__ancestor,
           match _0 with
           | vardef_0__ancestor_vertex => whatever2
           | vardef_0__ancestor_work => a
           end) vardef_0__ancestor_work 
        (Z.of_nat g))
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
    assert (step3 : (update
    (λ _0 : varsfuncdef_0__ancestor,
       match _0 with
       | vardef_0__ancestor_vertex => whatever2
       | vardef_0__ancestor_work => a
       end) vardef_0__ancestor_work (Z.of_nat g)) = fun x => match x with | vardef_0__ancestor_vertex => whatever2 | vardef_0__ancestor_work => Z.of_nat g end).
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
update
  (λ _0 : varsfuncdef_0__ancestor,
     match _0 with
     | vardef_0__ancestor_vertex => whatever2
     | vardef_0__ancestor_work => a
     end) vardef_0__ancestor_work 
  (Z.of_nat g) =
λ _0 : varsfuncdef_0__ancestor,
  match _0 with
  | vardef_0__ancestor_vertex => whatever2
  | vardef_0__ancestor_work => Z.of_nat g
  end
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: update
  (λ _0 : varsfuncdef_0__ancestor,
     match _0 with
     | vardef_0__ancestor_vertex => whatever2
     | vardef_0__ancestor_work => a
     end) vardef_0__ancestor_work 
  (Z.of_nat g) =
λ _0 : varsfuncdef_0__ancestor,
  match _0 with
  | vardef_0__ancestor_vertex => whatever2
  | vardef_0__ancestor_work => Z.of_nat g
  end
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (update
        (λ _0 : varsfuncdef_0__ancestor,
           match _0 with
           | vardef_0__ancestor_vertex => whatever2
           | vardef_0__ancestor_work => a
           end) vardef_0__ancestor_work 
        (Z.of_nat g))
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) {
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
update
  (λ _0 : varsfuncdef_0__ancestor,
     match _0 with
     | vardef_0__ancestor_vertex => whatever2
     | vardef_0__ancestor_work => a
     end) vardef_0__ancestor_work 
  (Z.of_nat g) =
λ _0 : varsfuncdef_0__ancestor,
  match _0 with
  | vardef_0__ancestor_vertex => whatever2
  | vardef_0__ancestor_work => Z.of_nat g
  end unfold update.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
(λ _0 : varsfuncdef_0__ancestor,
   if decide (_0 = vardef_0__ancestor_work)
   then Z.of_nat g
   else
    match _0 with
    | vardef_0__ancestor_vertex => whatever2
    | vardef_0__ancestor_work => a
    end) =
λ _0 : varsfuncdef_0__ancestor,
  match _0 with
  | vardef_0__ancestor_vertex => whatever2
  | vardef_0__ancestor_work => Z.of_nat g
  end apply functional_extensionality_dep.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
∀ _0 : varsfuncdef_0__ancestor,
  (if decide (_0 = vardef_0__ancestor_work)
   then Z.of_nat g
   else
    match _0 with
    | vardef_0__ancestor_vertex => whatever2
    | vardef_0__ancestor_work => a
    end) =
  match _0 with
  | vardef_0__ancestor_vertex => whatever2
  | vardef_0__ancestor_work => Z.of_nat g
  end intro aa.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
aa: varsfuncdef_0__ancestor
(if decide (aa = vardef_0__ancestor_work)
 then Z.of_nat g
 else
  match aa with
  | vardef_0__ancestor_vertex => whatever2
  | vardef_0__ancestor_work => a
  end) =
match aa with
| vardef_0__ancestor_vertex => whatever2
| vardef_0__ancestor_work => Z.of_nat g
end destruct aa; easy. }
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: update
  (λ _0 : varsfuncdef_0__ancestor,
     match _0 with
     | vardef_0__ancestor_vertex => whatever2
     | vardef_0__ancestor_work => a
     end) vardef_0__ancestor_work 
  (Z.of_nat g) =
λ _0 : varsfuncdef_0__ancestor,
  match _0 with
  | vardef_0__ancestor_vertex => whatever2
  | vardef_0__ancestor_work => Z.of_nat g
  end
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (update
        (λ _0 : varsfuncdef_0__ancestor,
           match _0 with
           | vardef_0__ancestor_vertex => whatever2
           | vardef_0__ancestor_work => a
           end) vardef_0__ancestor_work 
        (Z.of_nat g))
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite step3.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: update
  (λ _0 : varsfuncdef_0__ancestor,
     match _0 with
     | vardef_0__ancestor_vertex => whatever2
     | vardef_0__ancestor_work => a
     end) vardef_0__ancestor_work 
  (Z.of_nat g) =
λ _0 : varsfuncdef_0__ancestor,
  match _0 with
  | vardef_0__ancestor_vertex => whatever2
  | vardef_0__ancestor_work => Z.of_nat g
  end
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => Z.of_nat g
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) clear step3.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => Z.of_nat g
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
    pose proof IH ltac:(lia) (Z.of_nat g) ltac:(pose proof h1 (Z.to_nat a) _ hg as ls; lia) ltac:(lia) as step3.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state1 state2
  communication 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex =>
            whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat g
        end)
     (λ _ : varsfuncdef_0__ancestor,
        repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0
                      environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work >>=
                 λ _0 : 
                    withLocalVariablesReturnValue
                      (NumberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                   Done
                     (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue
                     Z (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0
                       arraydef_0__dsu,
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool
                       (bool_decide (...)%Z)) >>=
              λ _0 : bool,
                (if _0
                 then
                  break arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue ()
                      ()
                 else
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue ()
                    ()) >>=
                λ _ : (),
                  liftToWithinLoop
                    (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                     λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       numberLocalSet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work
                       _3) >>=
                  λ _ : (),
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue ()
                      ())) >>= continuation) >>=
   continuation2) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state1 state2
  communication 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex =>
            whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat
              (ancestor dsu n
                 (Z.to_nat (Z.of_nat g)))
        end)
     (λ _ : varsfuncdef_0__ancestor,
        repeat 0%Z 20) (continuation ()) >>=
   continuation2)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => Z.of_nat g
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite !liftToWithinLoopBind, <- !bindAssoc in step3.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state1 state2
  communication 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex =>
            whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat g
        end)
     (λ _ : varsfuncdef_0__ancestor,
        repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0
                      environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue
                     Z (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0
                       arraydef_0__dsu,
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue
                       () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue ()
                      ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables
                       arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0
                       ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>=
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue
                       () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state1 state2
  communication 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex =>
            whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat
              (ancestor dsu n
                 (Z.to_nat (Z.of_nat g)))
        end)
     (λ _ : varsfuncdef_0__ancestor,
        repeat 0%Z 20) (continuation ()) >>=
   continuation2)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => Z.of_nat g
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables arrayIndex0
                       (...) varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool 
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue () ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables arrayIndex0
                       ... varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>= 
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite step3, Nat2Z.id.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state1 state2
  communication 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex =>
            whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat g
        end)
     (λ _ : varsfuncdef_0__ancestor,
        repeat 0%Z 20)
     (loop n
        (λ _ : nat,
           dropWithinLoop
             (liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0
                      environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _0 : withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue
                     Z (coerceInt _0 64) >>=
                   λ _1 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _1 >>=
                     λ _2 : 
                       arrayType arrayIndex0
                       environment0
                       arraydef_0__dsu,
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt 0 8) >>=
                       λ _3 : Z,
                       Done
                       (WithLocalVariables
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       bool
                       (bool_decide (...)%Z)) >>=
                λ _1 : bool,
                  (if _1
                   then
                    break arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor >>=
                    λ _ : (),
                      Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue
                       () ()
                   else
                    Done
                      (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                      withinLoopReturnValue ()
                      ()) >>=
                  λ _ : (),
                    liftToWithinLoop
                      (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                    λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                      liftToWithinLoop
                       (Done
                       (WithLocalVariables
                       arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _2 64) >>=
                       λ _3 : Z,
                       retrieve arrayIndex0
                       ...
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _3 >>=
                       ..., ...) >>=
                      λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue
                       () ())) >>=
      continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20)
  (repeat 0%Z 20) 0 state1 state2
  communication 1 arrayIndex0
  arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list
            (arrayType arrayIndex0
               environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main
     (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex =>
            whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat
              (ancestor dsu n
                 (Z.to_nat (Z.of_nat g)))
        end)
     (λ _ : varsfuncdef_0__ancestor,
        repeat 0%Z 20) (continuation ()) >>=
   continuation2)
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu n g)
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
    clear step3.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu n g)
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) assert (step3 : ancestor dsu (S n) (Z.to_nat a) = ancestor dsu n g).
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
ancestor dsu (S n) (Z.to_nat a) = ancestor dsu n g
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: ancestor dsu (S n) (Z.to_nat a) =
ancestor dsu n g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu n g)
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) {
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
ancestor dsu (S n) (Z.to_nat a) = ancestor dsu n g simpl.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
match nth (Z.to_nat a) dsu (Ancestor Unit) with
| ReferTo _0 => ancestor dsu n _0
| Ancestor _ => Z.to_nat a
end = ancestor dsu n g rewrite hg.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
ancestor dsu n g = ancestor dsu n g reflexivity. }
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: ancestor dsu (S n) (Z.to_nat a) =
ancestor dsu n g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu n g)
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite step3.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: nat
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = ReferTo g
hs: Nat.lt (Z.to_nat a) (length (convertToArray dsu))
step3: ancestor dsu (S n) (Z.to_nat a) =
ancestor dsu n g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu n g)
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu n g)
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) reflexivity.
  +
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool
              (bool_decide
                 (toSigned
                    (256 - Z.of_nat (leafCount g)) 8 <
                  0)%Z)) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) case_bool_decide as hs.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool true) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: ¬ (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
    *
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool true) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite liftToWithinLoopBind, <- !bindAssoc, dropWithinLoopLiftToWithinLoop.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (Done
        (WithLocalVariables arrayIndex0
           (arrayType arrayIndex0 environment0)
           varsfuncdef_0__ancestor)
        withLocalVariablesReturnValue bool true >>=
      (λ _0 : bool,
         dropWithinLoop
           ((if _0
             then
              break arrayIndex0
                (arrayType arrayIndex0 environment0)
                varsfuncdef_0__ancestor >>=
              λ _ : (),
                Done
                  (WithinLoop arrayIndex0
                     (arrayType arrayIndex0
                       environment0)
                     varsfuncdef_0__ancestor)
                  withinLoopReturnValue () ()
             else
              Done
                (WithinLoop arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor)
                withinLoopReturnValue () ()) >>=
            λ _ : (),
              liftToWithinLoop
                (numberLocalGet arrayIndex0
                   (arrayType arrayIndex0 environment0)
                   varsfuncdef_0__ancestor
                   vardef_0__ancestor_work) >>=
              λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                liftToWithinLoop
                  (Done
                     (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                     withLocalVariablesReturnValue Z
                     (coerceInt _1 64) >>=
                   λ _2 : Z,
                     retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                     λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                       numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
                λ _ : (),
                  Done
                    (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withinLoopReturnValue () ())) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work) >>=
                       λ _2 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0 ...
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       liftToWithinLoop
                       (Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite <- !bindAssoc, leftIdentity, <- !bindAssoc, dropWithinLoop_break, !leftIdentity.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
      assert (step3 : ancestor dsu (S n) (Z.to_nat a) = Z.to_nat a).
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
ancestor dsu (S n) (Z.to_nat a) = Z.to_nat a
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
step3: ancestor dsu (S n) (Z.to_nat a) = Z.to_nat a
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
      {
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
ancestor dsu (S n) (Z.to_nat a) = Z.to_nat a simpl.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
match nth (Z.to_nat a) dsu (Ancestor Unit) with
| ReferTo _0 => ancestor dsu n _0
| Ancestor _ => Z.to_nat a
end = Z.to_nat a rewrite hg.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
Z.to_nat a = Z.to_nat a reflexivity. }
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
step3: ancestor dsu (S n) (Z.to_nat a) = Z.to_nat a
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite step3, Z2Nat.id; [| lia].
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
step3: ancestor dsu (S n) (Z.to_nat a) = Z.to_nat a
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) reflexivity.
    *
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: ¬ (toSigned (256 - Z.of_nat (leafCount g)) 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) unfold toSigned in hs.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hs: ¬ ((if
     decide
       (256 - Z.of_nat (leafCount g) <
        2 ^ (8 - 1))
    then 256 - Z.of_nat (leafCount g)
    else 256 - Z.of_nat (leafCount g) - 2 ^ 8) <
   0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) case_decide as hss.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hss: (256 - Z.of_nat (leafCount g) < 2 ^ (8 - 1))%Z
hs: ¬ (256 - Z.of_nat (leafCount g) < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2)
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hss: ¬ (256 - Z.of_nat (leafCount g) < 2 ^ (8 - 1))%Z
hs: ¬ (256 - Z.of_nat (leafCount g) - 2 ^ 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) {
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hss: (256 - Z.of_nat (leafCount g) < 2 ^ (8 - 1))%Z
hs: ¬ (256 - Z.of_nat (leafCount g) < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) simpl in hss.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hss: (256 - Z.of_nat (leafCount g) < 2 ^ (8 - 1))%Z
hs: ¬ (256 - Z.of_nat (leafCount g) < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite (ltac:(easy) : (2 ^ (8 - 1) = 128)%Z) in hss.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hss: (256 - Z.of_nat (leafCount g) < 128)%Z
hs: ¬ (256 - Z.of_nat (leafCount g) < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) pose proof nthLowerBoundConvertAuxStep dsu ltac:(lia) ltac:(lia) (Z.to_nat a) ltac:(lia) g hg.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hss: (256 - Z.of_nat (leafCount g) < 128)%Z
hs: ¬ (256 - Z.of_nat (leafCount g) < 0)%Z
_0: leafCount g < 128
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) lia. }
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hss: ¬ (256 - Z.of_nat (leafCount g) < 2 ^ (8 - 1))%Z
hs: ¬ (256 - Z.of_nat (leafCount g) - 2 ^ 8 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) rewrite (ltac:(easy) : (2 ^ 8 = 256)%Z) in hs.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hss: ¬ (256 - Z.of_nat (leafCount g) < 2 ^ (8 - 1))%Z
hs: ¬ (256 - Z.of_nat (leafCount g) - 256 < 0)%Z
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) pose proof oneLeqLeafCount g.
dsu: list Slot
state1, state2: BlockchainState
hL: length dsu = 100
hM: Z.to_nat (dsuLeafCount dsu) = length dsu
h1: noIllegalIndices dsu
h2: withoutCyclesN dsu (length dsu)
whatever2: Z
communication: list Z
continuation: ()
→ Action
    (WithLocalVariables arrayIndex0
       (arrayType arrayIndex0
          environment0)
       varsfuncdef_0__ancestor)
    withLocalVariablesReturnValue ()
continuation2: ()
→ Action
    (WithArrays arrayIndex0
       (arrayType arrayIndex0
          environment0))
    withArraysReturnValue ()
whatever: arrayType arrayIndex0 environment0
  arraydef_0__result
n: nat
hN: S n ≤ 100
IH: n ≤ 100
→ ∀ _0 : Z,
    (_0 < 100)%Z
    → (0 ≤ _0)%Z
      → invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work => _0
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20)
             (loop n
                (λ _ : nat,
                   dropWithinLoop
                     (liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet
                       arrayIndex0 
                       (...)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue
                       Z (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                      λ _1 : bool,
                       (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ()) >>=
                       λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet
                       arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue
                       Z ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue ()
                       ())) >>= continuation) >>=
           continuation2) =
        invokeContractAux 
          (repeat 1%Z 20) 
          (repeat 0%Z 20) 0 state1 state2
          communication 1 arrayIndex0
          arrayIndexEqualityDecidable0
          (arrayType arrayIndex0 environment0)
          (λ _1 : arrayIndex0,
             match
               _1 as _2
               return
                 (list
                    (arrayType arrayIndex0
                       environment0 _2))
             with
             | arraydef_0__dsu =>
                 convertToArray dsu
             | arraydef_0__hasBeenInitialized =>
                 [1%Z]
             | arraydef_0__result => [whatever]
             end)
          (funcdef_0__main
             (λ _ : varsfuncdef_0__main, false)
             (λ _ : varsfuncdef_0__main, 0%Z)
             (λ _ : varsfuncdef_0__main,
                repeat 0%Z 20))
          (eliminateLocalVariables
             (λ _ : varsfuncdef_0__ancestor,
                false)
             (λ _1 : varsfuncdef_0__ancestor,
                match _1 with
                | vardef_0__ancestor_vertex =>
                    whatever2
                | vardef_0__ancestor_work =>
                    Z.of_nat
                      (ancestor dsu n
                       (Z.to_nat _0))
                end)
             (λ _ : varsfuncdef_0__ancestor,
                repeat 0%Z 20) 
             (continuation ()) >>= continuation2)
a: Z
hLt1: (a < 100)%Z
hLe1: (0 ≤ a)%Z
step: coerceInt a 64 = a
g: Tree
hg: nth (Z.to_nat a) dsu (Ancestor Unit) = Ancestor g
hss: ¬ (256 - Z.of_nat (leafCount g) < 2 ^ (8 - 1))%Z
hs: ¬ (256 - Z.of_nat (leafCount g) - 256 < 0)%Z
_0: 1 ≤ leafCount g
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work => a
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (dropWithinLoop
        (liftToWithinLoop
           (Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue bool false) >>=
         λ _0 : bool,
           (if _0
            then
             break arrayIndex0
               (arrayType arrayIndex0 environment0)
               varsfuncdef_0__ancestor >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()
            else
             Done
               (WithinLoop arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor)
               withinLoopReturnValue () ()) >>=
           λ _ : (),
             liftToWithinLoop
               (numberLocalGet arrayIndex0
                  (arrayType arrayIndex0 environment0)
                  varsfuncdef_0__ancestor
                  vardef_0__ancestor_work >>=
                λ _1 : withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                  Done
                    (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                    withLocalVariablesReturnValue Z
                    (coerceInt _1 64) >>=
                  λ _2 : Z,
                    retrieve arrayIndex0
                      (arrayType arrayIndex0
                       environment0)
                      varsfuncdef_0__ancestor
                      arraydef_0__dsu _2 >>=
                    λ _3 : 
                       arrayType arrayIndex0
                       environment0 arraydef_0__dsu,
                      numberLocalSet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work _3) >>=
             λ _ : (),
               Done
                 (WithinLoop arrayIndex0
                    (arrayType arrayIndex0
                       environment0)
                    varsfuncdef_0__ancestor)
                 withinLoopReturnValue () ()) >>=
      λ _0 : LoopOutcome,
        match _0 with
        | KeepGoing =>
            loop n
              (λ _ : nat,
                 dropWithinLoop
                   (liftToWithinLoop
                      (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _1 : 
                       withLocalVariablesReturnValue
                       (NumberLocalGet arrayIndex0
                       (...) varsfuncdef_0__ancestor
                       vardef_0__ancestor_work),
                       Done
                       (WithLocalVariables arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withLocalVariablesReturnValue Z
                       (coerceInt _1 64) >>=
                       λ _2 : Z,
                       retrieve arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       arraydef_0__dsu _2 >>=
                       λ _3 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                    λ _1 : bool,
                      (if _1
                       then
                       break arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0 ...
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()
                       else
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()) >>=
                      λ _ : (),
                       liftToWithinLoop
                       (numberLocalGet arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor
                       vardef_0__ancestor_work >>=
                       λ _2 : ...,
                       Done ...
                       withLocalVariablesReturnValue Z
                       ... >>= 
                       ..., ...) >>=
                       λ _ : (),
                       Done
                       (WithinLoop arrayIndex0
                       (arrayType arrayIndex0
                       environment0)
                       varsfuncdef_0__ancestor)
                       withinLoopReturnValue () ()))
        | Stop =>
            Done
              (WithLocalVariables arrayIndex0
                 (arrayType arrayIndex0 environment0)
                 varsfuncdef_0__ancestor)
              withLocalVariablesReturnValue () ()
        end >>= continuation) >>= continuation2) =
invokeContractAux (repeat 1%Z 20) 
  (repeat 0%Z 20) 0 state1 state2 communication 1
  arrayIndex0 arrayIndexEqualityDecidable0
  (arrayType arrayIndex0 environment0)
  (λ _0 : arrayIndex0,
     match
       _0 as _1
       return
         (list (arrayType arrayIndex0 environment0 _1))
     with
     | arraydef_0__dsu => convertToArray dsu
     | arraydef_0__hasBeenInitialized => [1%Z]
     | arraydef_0__result => [whatever]
     end)
  (funcdef_0__main (λ _ : varsfuncdef_0__main, false)
     (λ _ : varsfuncdef_0__main, 0%Z)
     (λ _ : varsfuncdef_0__main, repeat 0%Z 20))
  (eliminateLocalVariables
     (λ _ : varsfuncdef_0__ancestor, false)
     (λ _0 : varsfuncdef_0__ancestor,
        match _0 with
        | vardef_0__ancestor_vertex => whatever2
        | vardef_0__ancestor_work =>
            Z.of_nat (ancestor dsu (S n) (Z.to_nat a))
        end)
     (λ _ : varsfuncdef_0__ancestor, repeat 0%Z 20)
     (continuation ()) >>= continuation2) lia.
Qed.

Fixpoint roll (dsu : list Slot) : option Tree :=
  match dsu with
  | [] => None
  | Ancestor x :: rest =>
    match roll rest with
    | None => Some x
    | Some y => Some (Unite x y)
    end
  | ReferTo x :: rest => roll rest
  end.

Lemma maxScore (dsu : list Slot) : Z.to_nat (dsuScore dsu) <= score (default Unit (roll dsu)).
dsu: list Slot
Z.to_nat (dsuScore dsu)
≤ score (default Unit (roll dsu))
Proof.
dsu: list Slot
Z.to_nat (dsuScore dsu)
≤ score (default Unit (roll dsu))
  induction dsu as [| head tail IH].
Z.to_nat (dsuScore [])
≤ score (default Unit (roll []))
head: Slot
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
Z.to_nat (dsuScore (head :: tail))
≤ score (default Unit (roll (head :: tail))) {
Z.to_nat (dsuScore [])
≤ score (default Unit (roll [])) easy. }
head: Slot
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
Z.to_nat (dsuScore (head :: tail))
≤ score (default Unit (roll (head :: tail)))
  unfold dsuScore.
head: Slot
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
Z.to_nat
  (Z.of_nat
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => score _1
              end) (head :: tail))))
≤ score (default Unit (roll (head :: tail))) rewrite Nat2Z.id.
head: Slot
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) (head :: tail))
≤ score (default Unit (roll (head :: tail))) rewrite -> (ltac:(listsEqual) : head :: tail = [head] ++ tail) at 1.
head: Slot
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) ([head] ++ tail))
≤ score (default Unit (roll (head :: tail))) rewrite map_app, list_sum_app.
head: Slot
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [head]) +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ score (default Unit (roll (head :: tail)))
  destruct head as [x | x].
x: nat
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo x]) +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ score (default Unit (roll (ReferTo x :: tail)))
x: Tree
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [Ancestor x]) +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ score (default Unit (roll (Ancestor x :: tail)))
  -
x: nat
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [ReferTo x]) +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ score (default Unit (roll (ReferTo x :: tail))) simpl.
x: nat
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail) ≤ score (default Unit (roll tail)) unfold dsuScore in IH.
x: nat
tail: list Slot
IH: Z.to_nat
  (Z.of_nat
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => score _1
              end) tail)))
≤ score (default Unit (roll tail))
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail) ≤ score (default Unit (roll tail)) rewrite Nat2Z.id in IH.
x: nat
tail: list Slot
IH: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ score (default Unit (roll tail))
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail) ≤ score (default Unit (roll tail)) exact IH.
  -
x: Tree
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) [Ancestor x]) +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ score (default Unit (roll (Ancestor x :: tail))) simpl.
x: Tree
tail: list Slot
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (roll tail))
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ score
    (default Unit
       match roll tail with
       | Some _0 => Some (Unite x _0)
       | None => Some x
       end) remember (roll tail) as g eqn:hg.
x: Tree
tail: list Slot
g: option Tree
hg: g = roll tail
IH: Z.to_nat (dsuScore tail) ≤ score (default Unit g)
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ score
    (default Unit
       match g with
       | Some _0 => Some (Unite x _0)
       | None => Some x
       end) destruct g as [g |].
x: Tree
tail: list Slot
g: Tree
hg: Some g = roll tail
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (Some g))
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ score (default Unit (Some (Unite x g)))
x: Tree
tail: list Slot
hg: None = roll tail
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit None)
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail) ≤ score (default Unit (Some x))
    {
x: Tree
tail: list Slot
g: Tree
hg: Some g = roll tail
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit (Some g))
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ score (default Unit (Some (Unite x g))) simpl in *.
x: Tree
tail: list Slot
g: Tree
hg: Some g = roll tail
IH: Z.to_nat (dsuScore tail) ≤ score g
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ leafCount x + leafCount g + score x + score g unfold dsuScore in IH.
x: Tree
tail: list Slot
g: Tree
hg: Some g = roll tail
IH: Z.to_nat
  (Z.of_nat
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => score _1
              end) tail))) ≤ 
score g
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ leafCount x + leafCount g + score x + score g rewrite Nat2Z.id in IH.
x: Tree
tail: list Slot
g: Tree
hg: Some g = roll tail
IH: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail) ≤ score g
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail)
≤ leafCount x + leafCount g + score x + score g lia. }
x: Tree
tail: list Slot
hg: None = roll tail
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit None)
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail) ≤ score (default Unit (Some x))
    {
x: Tree
tail: list Slot
hg: None = roll tail
IH: Z.to_nat (dsuScore tail)
≤ score (default Unit None)
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail) ≤ score (default Unit (Some x)) simpl in *.
x: Tree
tail: list Slot
hg: None = roll tail
IH: Z.to_nat (dsuScore tail) ≤ 0
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail) ≤ score x unfold dsuScore in IH.
x: Tree
tail: list Slot
hg: None = roll tail
IH: Z.to_nat
  (Z.of_nat
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => score _1
              end) tail))) ≤ 0
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail) ≤ score x rewrite Nat2Z.id in IH.
x: Tree
tail: list Slot
hg: None = roll tail
IH: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail) ≤ 0
score x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => score _1
        end) tail) ≤ score x lia. }
Qed.

Lemma rollPreservesLeafCount (dsu : list Slot) : Z.to_nat (dsuLeafCount dsu) = match roll dsu with | None => 0 | Some x => leafCount x end.
dsu: list Slot
Z.to_nat (dsuLeafCount dsu) =
match roll dsu with
| Some _0 => leafCount _0
| None => 0
end
Proof.
dsu: list Slot
Z.to_nat (dsuLeafCount dsu) =
match roll dsu with
| Some _0 => leafCount _0
| None => 0
end
  induction dsu as [| head tail IH].
Z.to_nat (dsuLeafCount []) =
match roll [] with
| Some _0 => leafCount _0
| None => 0
end
head: Slot
tail: list Slot
IH: Z.to_nat (dsuLeafCount tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
Z.to_nat (dsuLeafCount (head :: tail)) =
match roll (head :: tail) with
| Some _0 => leafCount _0
| None => 0
end {
Z.to_nat (dsuLeafCount []) =
match roll [] with
| Some _0 => leafCount _0
| None => 0
end easy. }
head: Slot
tail: list Slot
IH: Z.to_nat (dsuLeafCount tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
Z.to_nat (dsuLeafCount (head :: tail)) =
match roll (head :: tail) with
| Some _0 => leafCount _0
| None => 0
end
  unfold dsuLeafCount.
head: Slot
tail: list Slot
IH: Z.to_nat (dsuLeafCount tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
Z.to_nat
  (Z.of_nat
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (head :: tail)))) =
match roll (head :: tail) with
| Some _0 => leafCount _0
| None => 0
end rewrite Nat2Z.id.
head: Slot
tail: list Slot
IH: Z.to_nat (dsuLeafCount tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) (head :: tail)) =
match roll (head :: tail) with
| Some _0 => leafCount _0
| None => 0
end rewrite -> (ltac:(listsEqual) : head :: tail = [head] ++ tail) at 1.
head: Slot
tail: list Slot
IH: Z.to_nat (dsuLeafCount tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) ([head] ++ tail)) =
match roll (head :: tail) with
| Some _0 => leafCount _0
| None => 0
end rewrite map_app, list_sum_app.
head: Slot
tail: list Slot
IH: Z.to_nat (dsuLeafCount tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) [head]) +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) =
match roll (head :: tail) with
| Some _0 => leafCount _0
| None => 0
end
  destruct head as [x | x]; simpl in *.
x: nat
tail: list Slot
IH: Z.to_nat (dsuLeafCount tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
x: Tree
tail: list Slot
IH: Z.to_nat (dsuLeafCount tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) =
match
  match roll tail with
  | Some _0 => Some (Unite x _0)
  | None => Some x
  end
with
| Some _0 => leafCount _0
| None => 0
end
  -
x: nat
tail: list Slot
IH: Z.to_nat (dsuLeafCount tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end unfold dsuLeafCount in IH.
x: nat
tail: list Slot
IH: Z.to_nat
  (Z.of_nat
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) tail))) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end rewrite Nat2Z.id in IH.
x: nat
tail: list Slot
IH: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end exact IH.
  -
x: Tree
tail: list Slot
IH: Z.to_nat (dsuLeafCount tail) =
match roll tail with
| Some _0 => leafCount _0
| None => 0
end
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) =
match
  match roll tail with
  | Some _0 => Some (Unite x _0)
  | None => Some x
  end
with
| Some _0 => leafCount _0
| None => 0
end remember (roll tail) as g eqn:hg.
x: Tree
tail: list Slot
g: option Tree
hg: g = roll tail
IH: Z.to_nat (dsuLeafCount tail) =
match g with
| Some _0 => leafCount _0
| None => 0
end
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) =
match
  match g with
  | Some _0 => Some (Unite x _0)
  | None => Some x
  end
with
| Some _0 => leafCount _0
| None => 0
end destruct g as [g |].
x: Tree
tail: list Slot
g: Tree
hg: Some g = roll tail
IH: Z.to_nat (dsuLeafCount tail) = leafCount g
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount (Unite x g)
x: Tree
tail: list Slot
hg: None = roll tail
IH: Z.to_nat (dsuLeafCount tail) = 0
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount x
    {
x: Tree
tail: list Slot
g: Tree
hg: Some g = roll tail
IH: Z.to_nat (dsuLeafCount tail) = leafCount g
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount (Unite x g) simpl in *.
x: Tree
tail: list Slot
g: Tree
hg: Some g = roll tail
IH: Z.to_nat (dsuLeafCount tail) = leafCount g
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount x + leafCount g unfold dsuLeafCount in IH.
x: Tree
tail: list Slot
g: Tree
hg: Some g = roll tail
IH: Z.to_nat
  (Z.of_nat
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) tail))) = 
leafCount g
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount x + leafCount g rewrite Nat2Z.id in IH.
x: Tree
tail: list Slot
g: Tree
hg: Some g = roll tail
IH: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount g
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount x + leafCount g lia. }
x: Tree
tail: list Slot
hg: None = roll tail
IH: Z.to_nat (dsuLeafCount tail) = 0
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount x
    {
x: Tree
tail: list Slot
hg: None = roll tail
IH: Z.to_nat (dsuLeafCount tail) = 0
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount x simpl in *.
x: Tree
tail: list Slot
hg: None = roll tail
IH: Z.to_nat (dsuLeafCount tail) = 0
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount x unfold dsuLeafCount in IH.
x: Tree
tail: list Slot
hg: None = roll tail
IH: Z.to_nat
  (Z.of_nat
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) tail))) = 0
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount x rewrite Nat2Z.id in IH.
x: Tree
tail: list Slot
hg: None = roll tail
IH: list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = 0
leafCount x + 0 +
list_sum
  (map
     (λ _0 : Slot,
        match _0 with
        | ReferTo _ => 0
        | Ancestor _1 => leafCount _1
        end) tail) = leafCount x lia. }
Qed.

Lemma rollNoneScore (dsu : list Slot) (h : roll dsu = None) : dsuScore dsu = 0%Z.
dsu: list Slot
h: roll dsu = None
dsuScore dsu = 0%Z
Proof.
dsu: list Slot
h: roll dsu = None
dsuScore dsu = 0%Z
  induction dsu as [| head tail IH].
h: roll [] = None
dsuScore [] = 0%Z
head: Slot
tail: list Slot
h: roll (head :: tail) = None
IH: roll tail = None → dsuScore tail = 0%Z
dsuScore (head :: tail) = 0%Z {
h: roll [] = None
dsuScore [] = 0%Z easy. }
head: Slot
tail: list Slot
h: roll (head :: tail) = None
IH: roll tail = None → dsuScore tail = 0%Z
dsuScore (head :: tail) = 0%Z
  destruct head as [g |].
g: nat
tail: list Slot
h: roll (ReferTo g :: tail) = None
IH: roll tail = None → dsuScore tail = 0%Z
dsuScore (ReferTo g :: tail) = 0%Z
_0: Tree
tail: list Slot
h: roll (Ancestor _0 :: tail) = None
IH: roll tail = None → dsuScore tail = 0%Z
dsuScore (Ancestor _0 :: tail) = 0%Z
  -
g: nat
tail: list Slot
h: roll (ReferTo g :: tail) = None
IH: roll tail = None → dsuScore tail = 0%Z
dsuScore (ReferTo g :: tail) = 0%Z simpl in h.
g: nat
tail: list Slot
h: roll tail = None
IH: roll tail = None → dsuScore tail = 0%Z
dsuScore (ReferTo g :: tail) = 0%Z unfold dsuScore.
g: nat
tail: list Slot
h: roll tail = None
IH: roll tail = None → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) (ReferTo g :: tail))) = 0%Z rewrite (ltac:(easy) : ReferTo g :: tail = [ReferTo g] ++ tail).
g: nat
tail: list Slot
h: roll tail = None
IH: roll tail = None → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) ([ReferTo g] ++ tail))) = 0%Z rewrite map_app, list_sum_app.
g: nat
tail: list Slot
h: roll tail = None
IH: roll tail = None → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) [ReferTo g]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) tail)) = 0%Z simpl.
g: nat
tail: list Slot
h: roll tail = None
IH: roll tail = None → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) tail)) = 0%Z exact (IH h).
  -
_0: Tree
tail: list Slot
h: roll (Ancestor _0 :: tail) = None
IH: roll tail = None → dsuScore tail = 0%Z
dsuScore (Ancestor _0 :: tail) = 0%Z simpl in h.
_0: Tree
tail: list Slot
h: match roll tail with
| Some _1 => Some (Unite _0 _1)
| None => Some _0
end = None
IH: roll tail = None → dsuScore tail = 0%Z
dsuScore (Ancestor _0 :: tail) = 0%Z destruct (roll tail); easy.
Qed.

Lemma rollNoneLeafCount (dsu : list Slot) (h : roll dsu = None) : dsuLeafCount dsu = 0%Z.
dsu: list Slot
h: roll dsu = None
dsuLeafCount dsu = 0%Z
Proof.
dsu: list Slot
h: roll dsu = None
dsuLeafCount dsu = 0%Z
  induction dsu as [| head tail IH].
h: roll [] = None
dsuLeafCount [] = 0%Z
head: Slot
tail: list Slot
h: roll (head :: tail) = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
dsuLeafCount (head :: tail) = 0%Z {
h: roll [] = None
dsuLeafCount [] = 0%Z easy. }
head: Slot
tail: list Slot
h: roll (head :: tail) = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
dsuLeafCount (head :: tail) = 0%Z
  destruct head as [g |].
g: nat
tail: list Slot
h: roll (ReferTo g :: tail) = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
dsuLeafCount (ReferTo g :: tail) = 0%Z
_0: Tree
tail: list Slot
h: roll (Ancestor _0 :: tail) = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
dsuLeafCount (Ancestor _0 :: tail) = 0%Z
  -
g: nat
tail: list Slot
h: roll (ReferTo g :: tail) = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
dsuLeafCount (ReferTo g :: tail) = 0%Z simpl in h.
g: nat
tail: list Slot
h: roll tail = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
dsuLeafCount (ReferTo g :: tail) = 0%Z unfold dsuLeafCount.
g: nat
tail: list Slot
h: roll tail = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (ReferTo g :: tail))) = 0%Z rewrite (ltac:(easy) : ReferTo g :: tail = [ReferTo g] ++ tail).
g: nat
tail: list Slot
h: roll tail = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) ([ReferTo g] ++ tail))) = 0%Z rewrite map_app, list_sum_app.
g: nat
tail: list Slot
h: roll tail = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) [ReferTo g]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) tail)) = 0%Z simpl.
g: nat
tail: list Slot
h: roll tail = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) tail)) = 0%Z exact (IH h).
  -
_0: Tree
tail: list Slot
h: roll (Ancestor _0 :: tail) = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
dsuLeafCount (Ancestor _0 :: tail) = 0%Z simpl in h.
_0: Tree
tail: list Slot
h: match roll tail with
| Some _1 => Some (Unite _0 _1)
| None => Some _0
end = None
IH: roll tail = None → dsuLeafCount tail = 0%Z
dsuLeafCount (Ancestor _0 :: tail) = 0%Z destruct (roll tail); easy.
Qed.

Lemma maxScore2 (dsu : list Slot) : Z.to_nat (dsuScore dsu) <= score (constructTree (Z.to_nat (dsuLeafCount dsu) - 1)).
dsu: list Slot
Z.to_nat (dsuScore dsu)
≤ score
    (constructTree (Z.to_nat (dsuLeafCount dsu) - 1))
Proof.
dsu: list Slot
Z.to_nat (dsuScore dsu)
≤ score
    (constructTree (Z.to_nat (dsuLeafCount dsu) - 1))
  remember (roll dsu) as g eqn:hg.
dsu: list Slot
g: option Tree
hg: g = roll dsu
Z.to_nat (dsuScore dsu)
≤ score
    (constructTree (Z.to_nat (dsuLeafCount dsu) - 1)) destruct g as [g |]; symmetry in hg; [| rewrite rollNoneScore, rollNoneLeafCount; (assumption || lia)].
dsu: list Slot
g: Tree
hg: roll dsu = Some g
Z.to_nat (dsuScore dsu)
≤ score
    (constructTree (Z.to_nat (dsuLeafCount dsu) - 1))
  pose proof maxScore dsu as a1.
dsu: list Slot
g: Tree
hg: roll dsu = Some g
a1: Z.to_nat (dsuScore dsu)
≤ score (default Unit (roll dsu))
Z.to_nat (dsuScore dsu)
≤ score
    (constructTree (Z.to_nat (dsuLeafCount dsu) - 1))
  pose proof constructTreeIsOptimal (Z.to_nat (dsuLeafCount dsu) - 1) (default Unit (roll dsu)) as a2.
dsu: list Slot
g: Tree
hg: roll dsu = Some g
a1: Z.to_nat (dsuScore dsu)
≤ score (default Unit (roll dsu))
a2: leafCount
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
leafCount (default Unit (roll dsu))
→ score (default Unit (roll dsu))
  ≤ score
      (constructTree
         (Z.to_nat (dsuLeafCount dsu) - 1))
Z.to_nat (dsuScore dsu)
≤ score
    (constructTree (Z.to_nat (dsuLeafCount dsu) - 1))
  rewrite rollPreservesLeafCount in *.
dsu: list Slot
g: Tree
hg: roll dsu = Some g
a1: Z.to_nat (dsuScore dsu)
≤ score (default Unit (roll dsu))
a2: leafCount
  (constructTree
     (match roll dsu with
      | Some _0 => leafCount _0
      | None => 0
      end - 1)) =
leafCount (default Unit (roll dsu))
→ score (default Unit (roll dsu))
  ≤ score
      (constructTree
         (match roll dsu with
          | Some _0 => leafCount _0
          | None => 0
          end - 1))
Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (match roll dsu with
        | Some _0 => leafCount _0
        | None => 0
        end - 1)) rewrite hg in *.
dsu: list Slot
g: Tree
hg: roll dsu = Some g
a1: Z.to_nat (dsuScore dsu)
≤ score (default Unit (Some g))
a2: leafCount (constructTree (leafCount g - 1)) =
leafCount (default Unit (Some g))
→ score (default Unit (Some g))
  ≤ score (constructTree (leafCount g - 1))
Z.to_nat (dsuScore dsu)
≤ score (constructTree (leafCount g - 1)) simpl in *.
dsu: list Slot
g: Tree
hg: roll dsu = Some g
a1: Z.to_nat (dsuScore dsu) ≤ score g
a2: leafCount (constructTree (leafCount g - 1)) =
leafCount g
→ score g
  ≤ score (constructTree (leafCount g - 1))
Z.to_nat (dsuScore dsu)
≤ score (constructTree (leafCount g - 1))
  pose proof oneLeqLeafCount g.
dsu: list Slot
g: Tree
hg: roll dsu = Some g
a1: Z.to_nat (dsuScore dsu) ≤ score g
a2: leafCount (constructTree (leafCount g - 1)) =
leafCount g
→ score g
  ≤ score (constructTree (leafCount g - 1))
_0: 1 ≤ leafCount g
Z.to_nat (dsuScore dsu)
≤ score (constructTree (leafCount g - 1))
  pose proof a2 ltac:(rewrite constructTreeSize; lia).
dsu: list Slot
g: Tree
hg: roll dsu = Some g
a1: Z.to_nat (dsuScore dsu) ≤ score g
a2: leafCount (constructTree (leafCount g - 1)) =
leafCount g
→ score g
  ≤ score (constructTree (leafCount g - 1))
_0: 1 ≤ leafCount g
_1: score g ≤ score (constructTree (leafCount g - 1))
Z.to_nat (dsuScore dsu)
≤ score (constructTree (leafCount g - 1)) lia.
Qed.

Lemma scoreZeroIfLeafCountZero (dsu : list Slot) (h : dsuLeafCount dsu = 0%Z) : dsuScore dsu = 0%Z.
dsu: list Slot
h: dsuLeafCount dsu = 0%Z
dsuScore dsu = 0%Z
Proof.
dsu: list Slot
h: dsuLeafCount dsu = 0%Z
dsuScore dsu = 0%Z
  induction dsu as [| head tail IH].
h: dsuLeafCount [] = 0%Z
dsuScore [] = 0%Z
head: Slot
tail: list Slot
h: dsuLeafCount (head :: tail) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
dsuScore (head :: tail) = 0%Z {
h: dsuLeafCount [] = 0%Z
dsuScore [] = 0%Z easy. }
head: Slot
tail: list Slot
h: dsuLeafCount (head :: tail) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
dsuScore (head :: tail) = 0%Z
  unfold dsuScore.
head: Slot
tail: list Slot
h: dsuLeafCount (head :: tail) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) (head :: tail))) = 0%Z unfold dsuLeafCount in h.
head: Slot
tail: list Slot
h: Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (head :: tail))) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) (head :: tail))) = 0%Z rewrite (ltac:(easy) : head :: tail = [head] ++ tail), map_app, list_sum_app in *.
head: Slot
tail: list Slot
h: Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) [head]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) tail)) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) [head]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) tail)) = 0%Z
  destruct head as [x | x].
x: nat
tail: list Slot
h: Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) [ReferTo x]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) tail)) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) [ReferTo x]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) tail)) = 0%Z
x: Tree
tail: list Slot
h: Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) [Ancestor x]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) tail)) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) [Ancestor x]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) tail)) = 0%Z
  -
x: nat
tail: list Slot
h: Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) [ReferTo x]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) tail)) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) [ReferTo x]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) tail)) = 0%Z simpl in *.
x: nat
tail: list Slot
h: Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) tail)) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) tail)) = 0%Z exact (IH h).
  -
x: Tree
tail: list Slot
h: Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) [Ancestor x]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) tail)) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) [Ancestor x]) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) tail)) = 0%Z simpl in *.
x: Tree
tail: list Slot
h: Z.of_nat
  (leafCount x + 0 +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) tail)) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
Z.of_nat
  (score x + 0 +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) tail)) = 0%Z pose proof oneLeqLeafCount x.
x: Tree
tail: list Slot
h: Z.of_nat
  (leafCount x + 0 +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) tail)) = 0%Z
IH: dsuLeafCount tail = 0%Z → dsuScore tail = 0%Z
_0: 1 ≤ leafCount x
Z.of_nat
  (score x + 0 +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => score _1
           end) tail)) = 0%Z lia.
Qed.

Lemma maxScore3 (dsu : list Slot) : Z.to_nat (dsuScore dsu) <= (Z.to_nat (dsuLeafCount dsu)) * (Z.to_nat (dsuLeafCount dsu) + 1) / 2 - 1.
dsu: list Slot
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1
Proof.
dsu: list Slot
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1
  pose proof maxScore2 dsu as h1.
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1
  pose proof constructTreeScore (Z.to_nat (dsuLeafCount dsu) - 1) as h2.
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
((Z.to_nat (dsuLeafCount dsu) - 1 + 1) *
 (Z.to_nat (dsuLeafCount dsu) - 1 + 2)) `div` 2 -
1
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1
  assert (hS : (dsuLeafCount dsu = 0%Z \/ 0%Z < dsuLeafCount dsu)%Z).
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
((Z.to_nat (dsuLeafCount dsu) - 1 + 1) *
 (Z.to_nat (dsuLeafCount dsu) - 1 + 2)) `div` 2 -
1
dsuLeafCount dsu = 0%Z ∨ (0 < dsuLeafCount dsu)%Z
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
((Z.to_nat (dsuLeafCount dsu) - 1 + 1) *
 (Z.to_nat (dsuLeafCount dsu) - 1 + 2)) `div` 2 -
1
hS: dsuLeafCount dsu = 0%Z ∨ (0 < dsuLeafCount dsu)%Z
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1
  {
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
((Z.to_nat (dsuLeafCount dsu) - 1 + 1) *
 (Z.to_nat (dsuLeafCount dsu) - 1 + 2)) `div` 2 -
1
dsuLeafCount dsu = 0%Z ∨ (0 < dsuLeafCount dsu)%Z unfold dsuLeafCount.
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
((Z.to_nat (dsuLeafCount dsu) - 1 + 1) *
 (Z.to_nat (dsuLeafCount dsu) - 1 + 2)) `div` 2 -
1
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) dsu)) = 0%Z
∨ (0 <
   Z.of_nat
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0%nat
              | Ancestor _1 => leafCount _1
              end) dsu)))%Z lia. }
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
((Z.to_nat (dsuLeafCount dsu) - 1 + 1) *
 (Z.to_nat (dsuLeafCount dsu) - 1 + 2)) `div` 2 -
1
hS: dsuLeafCount dsu = 0%Z ∨ (0 < dsuLeafCount dsu)%Z
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1
  destruct hS as [hS | hS].
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
((Z.to_nat (dsuLeafCount dsu) - 1 + 1) *
 (Z.to_nat (dsuLeafCount dsu) - 1 + 2)) `div` 2 -
1
hS: dsuLeafCount dsu = 0%Z
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
((Z.to_nat (dsuLeafCount dsu) - 1 + 1) *
 (Z.to_nat (dsuLeafCount dsu) - 1 + 2)) `div` 2 -
1
hS: (0 < dsuLeafCount dsu)%Z
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1
  -
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
((Z.to_nat (dsuLeafCount dsu) - 1 + 1) *
 (Z.to_nat (dsuLeafCount dsu) - 1 + 2)) `div` 2 -
1
hS: dsuLeafCount dsu = 0%Z
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1 pose proof scoreZeroIfLeafCountZero _ hS as hI.
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
((Z.to_nat (dsuLeafCount dsu) - 1 + 1) *
 (Z.to_nat (dsuLeafCount dsu) - 1 + 2)) `div` 2 -
1
hS: dsuLeafCount dsu = 0%Z
hI: dsuScore dsu = 0%Z
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1 lia.
  -
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
((Z.to_nat (dsuLeafCount dsu) - 1 + 1) *
 (Z.to_nat (dsuLeafCount dsu) - 1 + 2)) `div` 2 -
1
hS: (0 < dsuLeafCount dsu)%Z
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1 rewrite (ltac:(lia) : Z.to_nat (dsuLeafCount dsu) - 1 + 1 = Z.to_nat (dsuLeafCount dsu)), (ltac:(lia) : Z.to_nat (dsuLeafCount dsu) - 1 + 2 = Z.to_nat (dsuLeafCount dsu) + 1) in h2.
dsu: list Slot
h1: Z.to_nat (dsuScore dsu)
≤ score
    (constructTree
       (Z.to_nat (dsuLeafCount dsu) - 1))
h2: score
  (constructTree
     (Z.to_nat (dsuLeafCount dsu) - 1)) =
(Z.to_nat (dsuLeafCount dsu) *
 (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1
hS: (0 < dsuLeafCount dsu)%Z
Z.to_nat (dsuScore dsu)
≤ (Z.to_nat (dsuLeafCount dsu) *
   (Z.to_nat (dsuLeafCount dsu) + 1)) `div` 2 - 1 lia.
Qed.

Lemma unitePreservesLength (dsu : list Slot) (a b : nat) : length (unite dsu a b) = length dsu.
dsu: list Slot
a, b: nat
length (unite dsu a b) = length dsu
Proof.
dsu: list Slot
a, b: nat
length (unite dsu a b) = length dsu
  unfold unite.
dsu: list Slot
a, b: nat
length
  (if decide (length dsu ≤ a)
   then dsu
   else
    if decide (length dsu ≤ b)
    then dsu
    else
     if
      decide
        (ancestor dsu (length dsu) a =
         ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
     then
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
     else
      match
        nth (ancestor dsu (length dsu) a)
          (pathCompress
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _0 =>
          match
            nth
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
              (pathCompress
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a (ancestor dsu (...) a))) b))
              (Ancestor Unit)
          with
          | ReferTo _ =>
              pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
          | Ancestor _1 =>
              if decide (leafCount _1 < leafCount _0)
              then
               performMerge
                 (pathCompress
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b
                    (ancestor
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a))
                       (length
                          (pathCompress dsu
                             (length dsu) a
                             (ancestor dsu (...) a)))
                       b)) _1 _0
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b) (ancestor dsu (length dsu) a)
              else
               performMerge
                 (pathCompress
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b
                    (ancestor
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a))
                       (length
                          (pathCompress dsu
                             (length dsu) a
                             (ancestor dsu (...) a)))
                       b)) _0 _1
                 (ancestor dsu (length dsu) a)
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b)
          end
      end) = length dsu case_decide as h1.
dsu: list Slot
a, b: nat
h1: length dsu ≤ a
length dsu = length dsu
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
length
  (if decide (length dsu ≤ b)
   then dsu
   else
    if
     decide
       (ancestor dsu (length dsu) a =
        ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    then
     pathCompress
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))) b
       (ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    else
     match
       nth (ancestor dsu (length dsu) a)
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         (Ancestor Unit)
     with
     | ReferTo _ =>
         pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)
     | Ancestor _0 =>
         match
           nth
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) (Ancestor Unit)
         with
         | ReferTo _ =>
             pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
         | Ancestor _1 =>
             if decide (leafCount _1 < leafCount _0)
             then
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _1 _0
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b) (ancestor dsu (length dsu) a)
             else
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _0 _1
                (ancestor dsu (length dsu) a)
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
         end
     end) = length dsu {
dsu: list Slot
a, b: nat
h1: length dsu ≤ a
length dsu = length dsu reflexivity. }
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
length
  (if decide (length dsu ≤ b)
   then dsu
   else
    if
     decide
       (ancestor dsu (length dsu) a =
        ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    then
     pathCompress
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))) b
       (ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    else
     match
       nth (ancestor dsu (length dsu) a)
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         (Ancestor Unit)
     with
     | ReferTo _ =>
         pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)
     | Ancestor _0 =>
         match
           nth
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) (Ancestor Unit)
         with
         | ReferTo _ =>
             pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
         | Ancestor _1 =>
             if decide (leafCount _1 < leafCount _0)
             then
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _1 _0
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b) (ancestor dsu (length dsu) a)
             else
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _0 _1
                (ancestor dsu (length dsu) a)
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
         end
     end) = length dsu case_decide as h2.
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: length dsu ≤ b
length dsu = length dsu
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
length
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu (length dsu) a
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _1 _0
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
        end
    end) = length dsu {
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: length dsu ≤ b
length dsu = length dsu reflexivity. }
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
length
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu (length dsu) a
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _1 _0
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
        end
    end) = length dsu case_decide as h3.
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))) b
length
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
length dsu
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
length
  match
    nth (ancestor dsu (length dsu) a)
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end = length dsu {
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))) b
length
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
length dsu rewrite !pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))) b
length dsu = length dsu reflexivity. }
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
length
  match
    nth (ancestor dsu (length dsu) a)
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end = length dsu
  destruct (nth _ _ _) as [m | m].
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: nat
length
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
length dsu
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
length
  match
    nth
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount m)
      then
       performMerge
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 m
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         m _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end = length dsu {
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: nat
length
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
length dsu rewrite !pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: nat
length dsu = length dsu reflexivity. }
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
length
  match
    nth
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount m)
      then
       performMerge
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 m
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         m _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end = length dsu destruct (nth _ _ _) as [n | n].
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
n: nat
length
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
length dsu
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m, n: Tree
length
  (if decide (leafCount n < leafCount m)
   then
    performMerge
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      n m
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (ancestor dsu (length dsu) a)
   else
    performMerge
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      m n (ancestor dsu (length dsu) a)
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)) =
length dsu {
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
n: nat
length
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
length dsu rewrite !pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
n: nat
length dsu = length dsu reflexivity. }
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m, n: Tree
length
  (if decide (leafCount n < leafCount m)
   then
    performMerge
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      n m
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (ancestor dsu (length dsu) a)
   else
    performMerge
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      m n (ancestor dsu (length dsu) a)
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)) =
length dsu case_decide as h4.
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m, n: Tree
h4: leafCount n < leafCount m
length
  (performMerge
     (pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)) n
     m
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)
     (ancestor dsu (length dsu) a)) = length dsu
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m, n: Tree
h4: ¬ leafCount n < leafCount m
length
  (performMerge
     (pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)) m
     n (ancestor dsu (length dsu) a)
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
length dsu
  -
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m, n: Tree
h4: leafCount n < leafCount m
length
  (performMerge
     (pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)) n
     m
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)
     (ancestor dsu (length dsu) a)) = length dsu unfold performMerge.
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m, n: Tree
h4: leafCount n < leafCount m
length
  (<[ancestor
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))) b:=ReferTo
                                                 (ancestor
                                                 dsu
                                                 (length
                                                 dsu)
                                                 a)]>
     (<[ancestor dsu (length dsu) a:=Ancestor
                                       (Unite m n)]>
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) =
length dsu rewrite !insert_length, !pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m, n: Tree
h4: leafCount n < leafCount m
length dsu = length dsu reflexivity.
  -
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m, n: Tree
h4: ¬ leafCount n < leafCount m
length
  (performMerge
     (pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)) m
     n (ancestor dsu (length dsu) a)
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
length dsu unfold performMerge.
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m, n: Tree
h4: ¬ leafCount n < leafCount m
length
  (<[ancestor dsu (length dsu) a:=ReferTo
                                    (ancestor
                                       (pathCompress
                                          dsu
                                          (length dsu)
                                          a
                                          (ancestor
                                             dsu
                                             (length
                                                dsu) a))
                                       (length
                                          (pathCompress
                                             dsu
                                             (length
                                                dsu) a
                                             (ancestor
                                                dsu
                                                (length
                                                 dsu)
                                                a))) b)]>
     (<[ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite n m)]>
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) =
length dsu rewrite !insert_length, !pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m, n: Tree
h4: ¬ leafCount n < leafCount m
length dsu = length dsu reflexivity.
Qed.

Lemma unitePreservesLeafCount (dsu : list Slot) (hN : noIllegalIndices dsu) (a b : nat) : dsuLeafCount (unite dsu a b) = dsuLeafCount dsu.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
dsuLeafCount (unite dsu a b) = dsuLeafCount dsu
Proof.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
dsuLeafCount (unite dsu a b) = dsuLeafCount dsu
  unfold unite.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
dsuLeafCount
  (if decide (length dsu ≤ a)
   then dsu
   else
    if decide (length dsu ≤ b)
    then dsu
    else
     if
      decide
        (ancestor dsu (length dsu) a =
         ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
     then
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
     else
      match
        nth (ancestor dsu (length dsu) a)
          (pathCompress
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _0 =>
          match
            nth
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
              (pathCompress
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a (ancestor dsu (...) a))) b))
              (Ancestor Unit)
          with
          | ReferTo _ =>
              pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
          | Ancestor _1 =>
              if decide (leafCount _1 < leafCount _0)
              then
               performMerge
                 (pathCompress
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b
                    (ancestor
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a))
                       (length
                          (pathCompress dsu
                             (length dsu) a
                             (ancestor dsu (...) a)))
                       b)) _1 _0
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b) (ancestor dsu (length dsu) a)
              else
               performMerge
                 (pathCompress
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b
                    (ancestor
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a))
                       (length
                          (pathCompress dsu
                             (length dsu) a
                             (ancestor dsu (...) a)))
                       b)) _0 _1
                 (ancestor dsu (length dsu) a)
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b)
          end
      end) = dsuLeafCount dsu case_decide as h1.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: length dsu ≤ a
dsuLeafCount dsu = dsuLeafCount dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
dsuLeafCount
  (if decide (length dsu ≤ b)
   then dsu
   else
    if
     decide
       (ancestor dsu (length dsu) a =
        ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    then
     pathCompress
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))) b
       (ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    else
     match
       nth (ancestor dsu (length dsu) a)
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         (Ancestor Unit)
     with
     | ReferTo _ =>
         pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)
     | Ancestor _0 =>
         match
           nth
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) (Ancestor Unit)
         with
         | ReferTo _ =>
             pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
         | Ancestor _1 =>
             if decide (leafCount _1 < leafCount _0)
             then
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _1 _0
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b) (ancestor dsu (length dsu) a)
             else
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _0 _1
                (ancestor dsu (length dsu) a)
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
         end
     end) = dsuLeafCount dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: length dsu ≤ a
dsuLeafCount dsu = dsuLeafCount dsu reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
dsuLeafCount
  (if decide (length dsu ≤ b)
   then dsu
   else
    if
     decide
       (ancestor dsu (length dsu) a =
        ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    then
     pathCompress
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))) b
       (ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    else
     match
       nth (ancestor dsu (length dsu) a)
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         (Ancestor Unit)
     with
     | ReferTo _ =>
         pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)
     | Ancestor _0 =>
         match
           nth
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) (Ancestor Unit)
         with
         | ReferTo _ =>
             pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
         | Ancestor _1 =>
             if decide (leafCount _1 < leafCount _0)
             then
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _1 _0
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b) (ancestor dsu (length dsu) a)
             else
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _0 _1
                (ancestor dsu (length dsu) a)
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
         end
     end) = dsuLeafCount dsu case_decide as h2.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: length dsu ≤ b
dsuLeafCount dsu = dsuLeafCount dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
dsuLeafCount
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu (length dsu) a
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _1 _0
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
        end
    end) = dsuLeafCount dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: length dsu ≤ b
dsuLeafCount dsu = dsuLeafCount dsu reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
dsuLeafCount
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu (length dsu) a
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _1 _0
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
        end
    end) = dsuLeafCount dsu case_decide as h3.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))) b
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
dsuLeafCount dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
dsuLeafCount
  match
    nth (ancestor dsu (length dsu) a)
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end = dsuLeafCount dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))) b
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
dsuLeafCount dsu rewrite !pathCompressPreservesLength, !pathCompressPreservesLeafCount.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))) b
dsuLeafCount dsu = dsuLeafCount dsu reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
dsuLeafCount
  match
    nth (ancestor dsu (length dsu) a)
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end = dsuLeafCount dsu
  remember (nth (ancestor dsu (length dsu) a) _ _) as ya eqn:yb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
ya: Slot
yb: ya =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount
  match ya with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end = dsuLeafCount dsu
  destruct ya as [m | m].
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: nat
yb: ReferTo m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
dsuLeafCount dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount
  match
    nth
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount m)
      then
       performMerge
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 m
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         m _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end = dsuLeafCount dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: nat
yb: ReferTo m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
dsuLeafCount dsu rewrite !pathCompressPreservesLength, !pathCompressPreservesLeafCount.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: nat
yb: ReferTo m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount dsu = dsuLeafCount dsu reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount
  match
    nth
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount m)
      then
       performMerge
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 m
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         m _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end = dsuLeafCount dsu remember (nth (ancestor (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
    (length (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)))
    b) _ _) as za eqn:zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
za: Slot
zb: za =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount
  match za with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount m)
      then
       performMerge
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 m
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         m _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end = dsuLeafCount dsu destruct za as [n | n].
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
n: nat
zb: ReferTo n =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
dsuLeafCount dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount
  (if decide (leafCount n < leafCount m)
   then
    performMerge
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      n m
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (ancestor dsu (length dsu) a)
   else
    performMerge
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      m n (ancestor dsu (length dsu) a)
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)) =
dsuLeafCount dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
n: nat
zb: ReferTo n =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
dsuLeafCount dsu rewrite !pathCompressPreservesLength, !pathCompressPreservesLeafCount.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
n: nat
zb: ReferTo n =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount dsu = dsuLeafCount dsu reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
dsuLeafCount
  (if decide (leafCount n < leafCount m)
   then
    performMerge
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      n m
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (ancestor dsu (length dsu) a)
   else
    performMerge
      (pathCompress
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      m n (ancestor dsu (length dsu) a)
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)) =
dsuLeafCount dsu case_decide as h4.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
dsuLeafCount
  (performMerge
     (pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)) n
     m
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)
     (ancestor dsu (length dsu) a)) = dsuLeafCount dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
dsuLeafCount
  (performMerge
     (pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)) m
     n (ancestor dsu (length dsu) a)
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
dsuLeafCount dsu
  -
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
dsuLeafCount
  (performMerge
     (pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)) n
     m
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)
     (ancestor dsu (length dsu) a)) = dsuLeafCount dsu unfold performMerge.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
dsuLeafCount
  (<[ancestor
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))) b:=ReferTo
                                                 (ancestor
                                                 dsu
                                                 (length
                                                 dsu)
                                                 a)]>
     (<[ancestor dsu (length dsu) a:=Ancestor
                                       (Unite m n)]>
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) =
dsuLeafCount dsu
    remember (ancestor dsu (length dsu) a) as x eqn:hx.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
dsuLeafCount
  (<[ancestor (pathCompress dsu (length dsu) a x)
       (length (pathCompress dsu (length dsu) a x)) b:=
     ReferTo x]>
     (<[x:=Ancestor (Unite m n)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b
           (ancestor
              (pathCompress dsu (length dsu) a x)
              (length
                 (pathCompress dsu (length dsu) a x))
              b)))) = dsuLeafCount dsu
    assert (xL : x < length dsu).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
x < length dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
dsuLeafCount
  (<[ancestor (pathCompress dsu (length dsu) a x)
       (length (pathCompress dsu (length dsu) a x)) b:=
     ReferTo x]>
     (<[x:=Ancestor (Unite m n)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b
           (ancestor
              (pathCompress dsu (length dsu) a x)
              (length
                 (pathCompress dsu (length dsu) a x))
              b)))) = dsuLeafCount dsu
    {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
x < length dsu rewrite hx.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
ancestor dsu (length dsu) a < length dsu apply ancestorLtLength.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
noIllegalIndices dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
a < length dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
noIllegalIndices dsu exact hN. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
a < length dsu lia. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
dsuLeafCount
  (<[ancestor (pathCompress dsu (length dsu) a x)
       (length (pathCompress dsu (length dsu) a x)) b:=
     ReferTo x]>
     (<[x:=Ancestor (Unite m n)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b
           (ancestor
              (pathCompress dsu (length dsu) a x)
              (length
                 (pathCompress dsu (length dsu) a x))
              b)))) = dsuLeafCount dsu
    remember (ancestor (pathCompress dsu (length dsu) a x) (length (pathCompress dsu (length dsu) a x)) b) as y eqn:hy.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
dsuLeafCount
  (<[y:=ReferTo x]>
     (<[x:=Ancestor (Unite m n)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b y))) = dsuLeafCount dsu
    assert (yL : y < length dsu).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
y < length dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
dsuLeafCount
  (<[y:=ReferTo x]>
     (<[x:=Ancestor (Unite m n)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b y))) = dsuLeafCount dsu
    {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
y < length dsu rewrite hy.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b <
length dsu rewrite (ltac:(rewrite pathCompressPreservesLength; reflexivity): length dsu = (length (pathCompress dsu (length dsu) a x))) at 3.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b <
length (pathCompress dsu (length dsu) a x) apply ancestorLtLength.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
noIllegalIndices (pathCompress dsu (length dsu) a x)
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
b < length (pathCompress dsu (length dsu) a x)
      -
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
noIllegalIndices (pathCompress dsu (length dsu) a x) apply pathCompressPreservesNoIllegalIndices.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
noIllegalIndices dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
a < length dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
x < length dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
noIllegalIndices dsu assumption. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
a < length dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
x < length dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
a < length dsu lia. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
x < length dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
x < length dsu exact xL. }
      -
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
b < length (pathCompress dsu (length dsu) a x) rewrite pathCompressPreservesLength.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
b < length dsu lia. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
dsuLeafCount
  (<[y:=ReferTo x]>
     (<[x:=Ancestor (Unite m n)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b y))) = dsuLeafCount dsu
    assert (lp : dsuLeafCount dsu = dsuLeafCount (pathCompress (pathCompress dsu (length dsu) a x) (length dsu) b y)).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
dsuLeafCount dsu =
dsuLeafCount
  (pathCompress (pathCompress dsu (length dsu) a x)
     (length dsu) b y)
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
lp: dsuLeafCount dsu =
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y)
dsuLeafCount
  (<[y:=ReferTo x]>
     (<[x:=Ancestor (Unite m n)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b y))) = dsuLeafCount dsu
    {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
dsuLeafCount dsu =
dsuLeafCount
  (pathCompress (pathCompress dsu (length dsu) a x)
     (length dsu) b y) rewrite !pathCompressPreservesLeafCount.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
dsuLeafCount dsu = dsuLeafCount dsu reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
lp: dsuLeafCount dsu =
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y)
dsuLeafCount
  (<[y:=ReferTo x]>
     (<[x:=Ancestor (Unite m n)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b y))) = dsuLeafCount dsu
    rewrite lp, !pathCompressPreservesLength.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
lp: dsuLeafCount dsu =
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y)
dsuLeafCount
  (<[y:=ReferTo x]>
     (<[x:=Ancestor (Unite m n)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length dsu) b y))) =
dsuLeafCount
  (pathCompress (pathCompress dsu (length dsu) a x)
     (length dsu) b y)
    remember (pathCompress (pathCompress dsu (length dsu) a x) (length dsu) b y) as f eqn:hf.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
dsuLeafCount
  (<[y:=ReferTo x]> (<[x:=Ancestor (Unite m n)]> f)) =
dsuLeafCount f
    assert (cr : length dsu = length f).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
length dsu = length f
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
dsuLeafCount
  (<[y:=ReferTo x]> (<[x:=Ancestor (Unite m n)]> f)) =
dsuLeafCount f
    {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
length dsu = length f rewrite hf, !pathCompressPreservesLength.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
length dsu = length dsu reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
dsuLeafCount
  (<[y:=ReferTo x]> (<[x:=Ancestor (Unite m n)]> f)) =
dsuLeafCount f
    destruct (ltac:(lia) : x < y \/ y < x) as [jj | jj].
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
dsuLeafCount
  (<[y:=ReferTo x]> (<[x:=Ancestor (Unite m n)]> f)) =
dsuLeafCount f
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
dsuLeafCount
  (<[y:=ReferTo x]> (<[x:=Ancestor (Unite m n)]> f)) =
dsuLeafCount f
    +
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
dsuLeafCount
  (<[y:=ReferTo x]> (<[x:=Ancestor (Unite m n)]> f)) =
dsuLeafCount f rewrite insert_take_drop, drop_insert_gt; try rewrite ?insert_length; try lia.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
dsuLeafCount
  (take y (<[x:=Ancestor (Unite m n)]> f) ++
   ReferTo x :: drop (S y) f) = dsuLeafCount f
      rewrite insert_take_drop, take_app, take_take, take_length, (ltac:(lia) : y `min` x = x), (ltac:(lia) : x `min` length f = x), (ltac:(easy) : Ancestor (Unite m n) :: drop (S x) f = [Ancestor (Unite m n)] ++ drop (S x) f), take_app; try lia.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
dsuLeafCount
  ((take x f ++
    take (y - x) [Ancestor (Unite m n)] ++
    take (y - x - length [Ancestor (Unite m n)])
      (drop (S x) f)) ++ ReferTo x :: drop (S y) f) =
dsuLeafCount f simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
dsuLeafCount
  ((take x f ++
    take (y - x) [Ancestor (Unite m n)] ++
    take (y - x - 1) (drop (S x) f)) ++
   ReferTo x :: drop (S y) f) = dsuLeafCount f
      assert (stp : take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
take (y - x) [Ancestor (Unite m n)] =
[Ancestor (Unite m n)]
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]
dsuLeafCount
  ((take x f ++
    take (y - x) [Ancestor (Unite m n)] ++
    take (y - x - 1) (drop (S x) f)) ++
   ReferTo x :: drop (S y) f) = dsuLeafCount f
      {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
take (y - x) [Ancestor (Unite m n)] =
[Ancestor (Unite m n)] rewrite (ltac:(lia) : y - x = S (y - x - 1)).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
take (S (y - x - 1)) [Ancestor (Unite m n)] =
[Ancestor (Unite m n)] simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
Ancestor (Unite m n) :: take (y - x - 1) [] =
[Ancestor (Unite m n)] rewrite take_nil.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
[Ancestor (Unite m n)] = [Ancestor (Unite m n)] reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]
dsuLeafCount
  ((take x f ++
    take (y - x) [Ancestor (Unite m n)] ++
    take (y - x - 1) (drop (S x) f)) ++
   ReferTo x :: drop (S y) f) = dsuLeafCount f
      rewrite stp, (ltac:(easy) : ReferTo x :: drop (S y) f = [ReferTo x] ++ drop (S y) f).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]
dsuLeafCount
  ((take x f ++
    [Ancestor (Unite m n)] ++
    take (y - x - 1) (drop (S x) f)) ++
   [ReferTo x] ++ drop (S y) f) = dsuLeafCount f
      rewrite (ListDecomposition.listDecomposition f x y jj ltac:(lia) (Ancestor Unit)) at 4.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]
dsuLeafCount
  ((take x f ++
    [Ancestor (Unite m n)] ++
    take (y - x - 1) (drop (S x) f)) ++
   [ReferTo x] ++ drop (S y) f) =
dsuLeafCount
  (take x f ++
   [nth x f (Ancestor Unit)] ++
   drop (S x) (take y f) ++
   [nth y f (Ancestor Unit)] ++ drop (y + 1) f) unfold dsuLeafCount.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end)
        ((take x f ++
          [Ancestor (Unite m n)] ++
          take (y - x - 1) (drop (S x) f)) ++
         [ReferTo x] ++ drop (S y) f))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end)
        (take x f ++
         [nth x f (Ancestor Unit)] ++
         drop (S x) (take y f) ++
         [nth y f (Ancestor Unit)] ++ drop (y + 1) f)))
      rewrite !map_app, !list_sum_app, take_drop_commute, (ltac:(lia) : S x + (y - x - 1) = y).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor (Unite m n)]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f)))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo x]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) f)))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth x f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth y f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (y + 1) f)))))) rewrite !pathCompressPreservesLength in yb, zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor (Unite m n)]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f)))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo x]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) f)))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth x f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth y f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (y + 1) f))))))
      rewrite <- hf in yb, zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor (Unite m n)]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f)))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo x]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) f)))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth x f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth y f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (y + 1) f)))))) rewrite <- yb, <- zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor (Unite m n)]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f)))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo x]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) f)))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor m]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor n]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (y + 1) f)))))) simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (leafCount m + leafCount n + 0 +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f)))) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S y) f))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (leafCount m + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (leafCount n + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (y + 1) f)))))) rewrite (ltac:(lia) : y + 1 = S y).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (y - x) [Ancestor (Unite m n)] = [Ancestor (Unite m n)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (leafCount m + leafCount n + 0 +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f)))) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S y) f))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (leafCount m + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (leafCount n + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S y) f)))))) lia.
    +
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
dsuLeafCount
  (<[y:=ReferTo x]> (<[x:=Ancestor (Unite m n)]> f)) =
dsuLeafCount f rewrite insert_take_drop, drop_insert_le; try rewrite ?insert_length; try lia.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
dsuLeafCount
  (take y (<[x:=Ancestor (Unite m n)]> f) ++
   ReferTo x
   :: <[x - S y:=Ancestor (Unite m n)]> (drop (S y) f)) =
dsuLeafCount f
      rewrite insert_take_drop, take_app, take_take, take_length, (ltac:(lia) : y `min` x = y), (ltac:(lia) : x `min` length f = x), (ltac:(easy) : Ancestor (Unite m n) :: drop (S x) f = [Ancestor (Unite m n)] ++ drop (S x) f), take_app; try lia.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
dsuLeafCount
  ((take y f ++
    take (y - x) [Ancestor (Unite m n)] ++
    take (y - x - length [Ancestor (Unite m n)])
      (drop (S x) f)) ++
   ReferTo x
   :: <[x - S y:=Ancestor (Unite m n)]> (drop (S y) f)) =
dsuLeafCount f simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
dsuLeafCount
  ((take y f ++
    take (y - x) [Ancestor (Unite m n)] ++
    take (y - x - 1) (drop (S x) f)) ++
   ReferTo x
   :: <[x - S y:=Ancestor (Unite m n)]> (drop (S y) f)) =
dsuLeafCount f
      assert (stp : take (y - x) [Ancestor (Unite m n)] = []).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
take (y - x) [Ancestor (Unite m n)] = []
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
dsuLeafCount
  ((take y f ++
    take (y - x) [Ancestor (Unite m n)] ++
    take (y - x - 1) (drop (S x) f)) ++
   ReferTo x
   :: <[x - S y:=Ancestor (Unite m n)]> (drop (S y) f)) =
dsuLeafCount f
      {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
take (y - x) [Ancestor (Unite m n)] = [] rewrite (ltac:(lia) : y - x = 0).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
take 0 [Ancestor (Unite m n)] = [] simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
[] = [] reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
dsuLeafCount
  ((take y f ++
    take (y - x) [Ancestor (Unite m n)] ++
    take (y - x - 1) (drop (S x) f)) ++
   ReferTo x
   :: <[x - S y:=Ancestor (Unite m n)]> (drop (S y) f)) =
dsuLeafCount f
      rewrite stp, !app_nil_l, insert_take_drop, drop_drop, !take_drop_commute, (ltac:(lia) : S y + S (x - S y) = S x), (ltac:(lia) : S x + (y - x - 1) = S x), (ltac:(lia) : S y + (x - S y) = x); [| rewrite drop_length; lia].
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
dsuLeafCount
  ((take y f ++ drop (S x) (take (S x) f)) ++
   ReferTo x
   :: drop (S y) (take x f) ++
      Ancestor (Unite m n) :: drop (S x) f) =
dsuLeafCount f
      rewrite (ltac:(intros; simpl; easy) : forall a b, a :: b = [a] ++ b), ((ltac:(intros; simpl; easy) : forall a b, a :: b = [a] ++ b) (Ancestor (Unite m n)) (drop (S x) f)).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
dsuLeafCount
  ((take y f ++ drop (S x) (take (S x) f)) ++
   [ReferTo x] ++
   drop (S y) (take x f) ++
   [Ancestor (Unite m n)] ++ drop (S x) f) =
dsuLeafCount f
      rewrite (ListDecomposition.listDecomposition f y x jj ltac:(lia) (Ancestor Unit)) at 5.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
dsuLeafCount
  ((take y f ++ drop (S x) (take (S x) f)) ++
   [ReferTo x] ++
   drop (S y) (take x f) ++
   [Ancestor (Unite m n)] ++ drop (S x) f) =
dsuLeafCount
  (take y f ++
   [nth y f (Ancestor Unit)] ++
   drop (S y) (take x f) ++
   [nth x f (Ancestor Unit)] ++ drop (x + 1) f) unfold dsuLeafCount.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end)
        ((take y f ++ drop (S x) (take (S x) f)) ++
         [ReferTo x] ++
         drop (S y) (take x f) ++
         [Ancestor (Unite m n)] ++ drop (S x) f))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end)
        (take y f ++
         [nth y f (Ancestor Unit)] ++
         drop (S y) (take x f) ++
         [nth x f (Ancestor Unit)] ++ drop (x + 1) f)))
      rewrite !map_app, !list_sum_app.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) (take (S x) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo x]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor (Unite m n)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f)))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth y f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth x f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (x + 1) f)))))) rewrite !pathCompressPreservesLength in yb, zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y) (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) (take (S x) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo x]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor (Unite m n)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f)))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth y f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth x f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (x + 1) f))))))
      rewrite <- hf in yb, zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) (take (S x) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo x]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor (Unite m n)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f)))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth y f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth x f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (x + 1) f)))))) rewrite <- yb, <- zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) (take (S x) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo x]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor (Unite m n)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f)))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor n]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor m]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (x + 1) f))))))
      simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) (take (S x) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f))) +
    (leafCount m + leafCount n + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (leafCount m + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (x + 1) f)))))) rewrite (ltac:(lia) : x + 1 = S x).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) (take (S x) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f))) +
    (leafCount m + leafCount n + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (leafCount m + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f))))))
      assert (ms : drop (S x) (take (S x) f) = []).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
drop (S x) (take (S x) f) = []
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
ms: drop (S x) (take (S x) f) = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) (take (S x) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f))) +
    (leafCount m + leafCount n + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (leafCount m + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f))))))
      {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
drop (S x) (take (S x) f) = [] assert (mt : length (drop (S x) (take (S x) f)) = 0).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
length (drop (S x) (take (S x) f)) = 0
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
mt: length (drop (S x) (take (S x) f)) = 0
drop (S x) (take (S x) f) = []
        {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
length (drop (S x) (take (S x) f)) = 0 rewrite drop_length, take_length.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
S x `min` length f - S x = 0 lia. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
mt: length (drop (S x) (take (S x) f)) = 0
drop (S x) (take (S x) f) = []
        exact (nil_length_inv _ mt). }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
ms: drop (S x) (take (S x) f) = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) (take (S x) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f))) +
    (leafCount m + leafCount n + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (leafCount m + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f))))))
      rewrite ms.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
ms: drop (S x) (take (S x) f) = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) []) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f))) +
    (leafCount m + leafCount n + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (leafCount m + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f)))))) simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (y - x) [Ancestor (Unite m n)] = []
ms: drop (S x) (take (S x) f) = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) + 0 +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f))) +
    (leafCount m + leafCount n + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (leafCount m + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f))))))
      lia.
  -
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
dsuLeafCount
  (performMerge
     (pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)) m
     n (ancestor dsu (length dsu) a)
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b)) =
dsuLeafCount dsu unfold performMerge.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
h3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))) b
m: Tree
yb: Ancestor m =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
dsuLeafCount
  (<[ancestor dsu (length dsu) a:=ReferTo
                                    (ancestor
                                       (pathCompress
                                          dsu
                                          (length dsu)
                                          a
                                          (ancestor
                                             dsu
                                             (length
                                                dsu) a))
                                       (length
                                          (pathCompress
                                             dsu
                                             (length
                                                dsu) a
                                             (ancestor
                                                dsu
                                                (length
                                                 dsu)
                                                a))) b)]>
     (<[ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite n m)]>
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) =
dsuLeafCount dsu
    remember (ancestor dsu (length dsu) a) as x eqn:hx.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
dsuLeafCount
  (<[x:=ReferTo
          (ancestor
             (pathCompress dsu (length dsu) a x)
             (length
                (pathCompress dsu (length dsu) a x)) b)]>
     (<[ancestor (pathCompress dsu (length dsu) a x)
          (length (pathCompress dsu (length dsu) a x))
          b:=Ancestor (Unite n m)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b
           (ancestor
              (pathCompress dsu (length dsu) a x)
              (length
                 (pathCompress dsu (length dsu) a x))
              b)))) = dsuLeafCount dsu
    assert (xL : x < length dsu).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
x < length dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
dsuLeafCount
  (<[x:=ReferTo
          (ancestor
             (pathCompress dsu (length dsu) a x)
             (length
                (pathCompress dsu (length dsu) a x)) b)]>
     (<[ancestor (pathCompress dsu (length dsu) a x)
          (length (pathCompress dsu (length dsu) a x))
          b:=Ancestor (Unite n m)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b
           (ancestor
              (pathCompress dsu (length dsu) a x)
              (length
                 (pathCompress dsu (length dsu) a x))
              b)))) = dsuLeafCount dsu
    {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
x < length dsu rewrite hx.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
ancestor dsu (length dsu) a < length dsu apply ancestorLtLength.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
noIllegalIndices dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
a < length dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
noIllegalIndices dsu exact hN. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
a < length dsu lia. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
h3: x
≠ ancestor (pathCompress dsu (length dsu) a x)
    (length (pathCompress dsu (length dsu) a x))
    b
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth
  (ancestor (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b)
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b
     (ancestor
        (pathCompress dsu (length dsu) a x)
        (length
           (pathCompress dsu (length dsu) a x)) b))
  (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
dsuLeafCount
  (<[x:=ReferTo
          (ancestor
             (pathCompress dsu (length dsu) a x)
             (length
                (pathCompress dsu (length dsu) a x)) b)]>
     (<[ancestor (pathCompress dsu (length dsu) a x)
          (length (pathCompress dsu (length dsu) a x))
          b:=Ancestor (Unite n m)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b
           (ancestor
              (pathCompress dsu (length dsu) a x)
              (length
                 (pathCompress dsu (length dsu) a x))
              b)))) = dsuLeafCount dsu
    remember (ancestor (pathCompress dsu (length dsu) a x) (length (pathCompress dsu (length dsu) a x)) b) as y eqn:hy.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
dsuLeafCount
  (<[x:=ReferTo y]>
     (<[y:=Ancestor (Unite n m)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b y))) = dsuLeafCount dsu
    assert (yL : y < length dsu).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
y < length dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
dsuLeafCount
  (<[x:=ReferTo y]>
     (<[y:=Ancestor (Unite n m)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b y))) = dsuLeafCount dsu
    {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
y < length dsu rewrite hy.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b <
length dsu rewrite (ltac:(rewrite pathCompressPreservesLength; reflexivity): length dsu = (length (pathCompress dsu (length dsu) a x))) at 3.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b <
length (pathCompress dsu (length dsu) a x) apply ancestorLtLength.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
noIllegalIndices (pathCompress dsu (length dsu) a x)
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
b < length (pathCompress dsu (length dsu) a x)
      -
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
noIllegalIndices (pathCompress dsu (length dsu) a x) apply pathCompressPreservesNoIllegalIndices.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
noIllegalIndices dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
a < length dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
x < length dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
noIllegalIndices dsu assumption. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
a < length dsu
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
x < length dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
a < length dsu lia. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
x < length dsu {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
x < length dsu exact xL. }
      -
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
b < length (pathCompress dsu (length dsu) a x) rewrite pathCompressPreservesLength.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
b < length dsu lia. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
dsuLeafCount
  (<[x:=ReferTo y]>
     (<[y:=Ancestor (Unite n m)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b y))) = dsuLeafCount dsu
    assert (lp : dsuLeafCount dsu = dsuLeafCount (pathCompress (pathCompress dsu (length dsu) a x) (length dsu) b y)).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
dsuLeafCount dsu =
dsuLeafCount
  (pathCompress (pathCompress dsu (length dsu) a x)
     (length dsu) b y)
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
lp: dsuLeafCount dsu =
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y)
dsuLeafCount
  (<[x:=ReferTo y]>
     (<[y:=Ancestor (Unite n m)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b y))) = dsuLeafCount dsu
    {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
dsuLeafCount dsu =
dsuLeafCount
  (pathCompress (pathCompress dsu (length dsu) a x)
     (length dsu) b y) rewrite !pathCompressPreservesLeafCount.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
dsuLeafCount dsu = dsuLeafCount dsu reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
lp: dsuLeafCount dsu =
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y)
dsuLeafCount
  (<[x:=ReferTo y]>
     (<[y:=Ancestor (Unite n m)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length (pathCompress dsu (length dsu) a x))
           b y))) = dsuLeafCount dsu
    rewrite lp, !pathCompressPreservesLength.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
lp: dsuLeafCount dsu =
dsuLeafCount
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y)
dsuLeafCount
  (<[x:=ReferTo y]>
     (<[y:=Ancestor (Unite n m)]>
        (pathCompress
           (pathCompress dsu (length dsu) a x)
           (length dsu) b y))) =
dsuLeafCount
  (pathCompress (pathCompress dsu (length dsu) a x)
     (length dsu) b y)
    remember (pathCompress (pathCompress dsu (length dsu) a x) (length dsu) b y) as f eqn:hf.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
dsuLeafCount
  (<[x:=ReferTo y]> (<[y:=Ancestor (Unite n m)]> f)) =
dsuLeafCount f
    assert (cr : length dsu = length f).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
length dsu = length f
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
dsuLeafCount
  (<[x:=ReferTo y]> (<[y:=Ancestor (Unite n m)]> f)) =
dsuLeafCount f
    {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
length dsu = length f rewrite hf, !pathCompressPreservesLength.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
length dsu = length dsu reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
dsuLeafCount
  (<[x:=ReferTo y]> (<[y:=Ancestor (Unite n m)]> f)) =
dsuLeafCount f
    destruct (ltac:(lia) : x < y \/ y < x) as [jj | jj].
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
dsuLeafCount
  (<[x:=ReferTo y]> (<[y:=Ancestor (Unite n m)]> f)) =
dsuLeafCount f
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
dsuLeafCount
  (<[x:=ReferTo y]> (<[y:=Ancestor (Unite n m)]> f)) =
dsuLeafCount f
    +
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
dsuLeafCount
  (<[x:=ReferTo y]> (<[y:=Ancestor (Unite n m)]> f)) =
dsuLeafCount f rewrite insert_take_drop, drop_insert_le; try rewrite ?insert_length; try lia.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
dsuLeafCount
  (take x (<[y:=Ancestor (Unite n m)]> f) ++
   ReferTo y
   :: <[y - S x:=Ancestor (Unite n m)]> (drop (S x) f)) =
dsuLeafCount f
      rewrite insert_take_drop, take_app, take_take, take_length, (ltac:(lia) : x `min` y = x), (ltac:(lia) : y `min` length f = y), (ltac:(easy) : Ancestor (Unite n m) :: drop (S y) f = [Ancestor (Unite n m)] ++ drop (S y) f), take_app; try lia.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
dsuLeafCount
  ((take x f ++
    take (x - y) [Ancestor (Unite n m)] ++
    take (x - y - length [Ancestor (Unite n m)])
      (drop (S y) f)) ++
   ReferTo y
   :: <[y - S x:=Ancestor (Unite n m)]> (drop (S x) f)) =
dsuLeafCount f simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
dsuLeafCount
  ((take x f ++
    take (x - y) [Ancestor (Unite n m)] ++
    take (x - y - 1) (drop (S y) f)) ++
   ReferTo y
   :: <[y - S x:=Ancestor (Unite n m)]> (drop (S x) f)) =
dsuLeafCount f
      assert (stp : take (x - y) [Ancestor (Unite n m)] = []).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
take (x - y) [Ancestor (Unite n m)] = []
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
dsuLeafCount
  ((take x f ++
    take (x - y) [Ancestor (Unite n m)] ++
    take (x - y - 1) (drop (S y) f)) ++
   ReferTo y
   :: <[y - S x:=Ancestor (Unite n m)]> (drop (S x) f)) =
dsuLeafCount f
      {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
take (x - y) [Ancestor (Unite n m)] = [] rewrite (ltac:(lia) : x - y = 0).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
take 0 [Ancestor (Unite n m)] = [] easy. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
dsuLeafCount
  ((take x f ++
    take (x - y) [Ancestor (Unite n m)] ++
    take (x - y - 1) (drop (S y) f)) ++
   ReferTo y
   :: <[y - S x:=Ancestor (Unite n m)]> (drop (S x) f)) =
dsuLeafCount f
      rewrite stp, !app_nil_l, insert_take_drop, drop_drop, !take_drop_commute, (ltac:(lia) : S x + S (y - S x) = S y), (ltac:(lia) : S x + (y - S x) = y), (ltac:(lia) : S y + (x - y - 1) = S y); [| rewrite drop_length; lia].
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
dsuLeafCount
  ((take x f ++ drop (S y) (take (S y) f)) ++
   ReferTo y
   :: drop (S x) (take y f) ++
      Ancestor (Unite n m) :: drop (S y) f) =
dsuLeafCount f
      rewrite (ltac:(intros; simpl; easy) : forall a b, a :: b = [a] ++ b), ((ltac:(intros; simpl; easy) : forall a b, a :: b = [a] ++ b) (Ancestor (Unite n m))).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
dsuLeafCount
  ((take x f ++ drop (S y) (take (S y) f)) ++
   [ReferTo y] ++
   drop (S x) (take y f) ++
   [Ancestor (Unite n m)] ++ drop (S y) f) =
dsuLeafCount f
      rewrite (ListDecomposition.listDecomposition f x y jj ltac:(lia) (Ancestor Unit)) at 5.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
dsuLeafCount
  ((take x f ++ drop (S y) (take (S y) f)) ++
   [ReferTo y] ++
   drop (S x) (take y f) ++
   [Ancestor (Unite n m)] ++ drop (S y) f) =
dsuLeafCount
  (take x f ++
   [nth x f (Ancestor Unit)] ++
   drop (S x) (take y f) ++
   [nth y f (Ancestor Unit)] ++ drop (y + 1) f) unfold dsuLeafCount.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end)
        ((take x f ++ drop (S y) (take (S y) f)) ++
         [ReferTo y] ++
         drop (S x) (take y f) ++
         [Ancestor (Unite n m)] ++ drop (S y) f))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end)
        (take x f ++
         [nth x f (Ancestor Unit)] ++
         drop (S x) (take y f) ++
         [nth y f (Ancestor Unit)] ++ drop (y + 1) f)))
      rewrite !map_app, !list_sum_app.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S y) (take (S y) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo y]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor (Unite n m)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S y) f)))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth x f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth y f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (y + 1) f)))))) rewrite !pathCompressPreservesLength in yb, zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S y) (take (S y) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo y]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor (Unite n m)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S y) f)))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth x f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth y f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (y + 1) f))))))
      rewrite <- hf in yb, zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S y) (take (S y) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo y]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor (Unite n m)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S y) f)))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth x f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth y f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (y + 1) f)))))) rewrite <- yb, <- zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S y) (take (S y) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo y]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor (Unite n m)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S y) f)))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor m]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor n]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (y + 1) f)))))) simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S y) (take (S y) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f))) +
    (leafCount n + leafCount m + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (leafCount m + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (leafCount n + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (y + 1) f)))))) rewrite (ltac:(lia) : y + 1 = S y).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S y) (take (S y) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f))) +
    (leafCount n + leafCount m + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (leafCount m + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (leafCount n + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S y) f))))))
      assert (ms : drop (S y) (take (S y) f) = []).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
drop (S y) (take (S y) f) = []
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
ms: drop (S y) (take (S y) f) = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S y) (take (S y) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f))) +
    (leafCount n + leafCount m + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (leafCount m + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (leafCount n + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S y) f))))))
      {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
drop (S y) (take (S y) f) = [] assert (mt : length (drop (S y) (take (S y) f)) = 0).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
length (drop (S y) (take (S y) f)) = 0
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
mt: length (drop (S y) (take (S y) f)) = 0
drop (S y) (take (S y) f) = []
        {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
length (drop (S y) (take (S y) f)) = 0 rewrite drop_length, take_length.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
S y `min` length f - S y = 0 lia. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
mt: length (drop (S y) (take (S y) f)) = 0
drop (S y) (take (S y) f) = []
        exact (nil_length_inv _ mt). }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
ms: drop (S y) (take (S y) f) = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S y) (take (S y) f))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f))) +
    (leafCount n + leafCount m + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (leafCount m + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (leafCount n + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S y) f))))))
      rewrite ms.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
ms: drop (S y) (take (S y) f) = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) []) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f))) +
    (leafCount n + leafCount m + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (leafCount m + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (leafCount n + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S y) f)))))) simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: x < y
stp: take (x - y) [Ancestor (Unite n m)] = []
ms: drop (S y) (take (S y) f) = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) + 0 +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) (take y f))) +
    (leafCount n + leafCount m + 0 +
     list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) f))))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take x f)) +
   (leafCount m + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S x) (take y f))) +
     (leafCount n + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S y) f))))))
      lia.
    +
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
dsuLeafCount
  (<[x:=ReferTo y]> (<[y:=Ancestor (Unite n m)]> f)) =
dsuLeafCount f rewrite insert_take_drop, drop_insert_gt; try rewrite ?insert_length; try lia.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
dsuLeafCount
  (take x (<[y:=Ancestor (Unite n m)]> f) ++
   ReferTo y :: drop (S x) f) = dsuLeafCount f
      rewrite insert_take_drop, take_app, take_take, take_length, (ltac:(lia) : x `min` y = y), (ltac:(lia) : y `min` length f = y), (ltac:(easy) : Ancestor (Unite n m) :: drop (S y) f = [Ancestor (Unite n m)] ++ drop (S y) f), take_app; try lia.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
dsuLeafCount
  ((take y f ++
    take (x - y) [Ancestor (Unite n m)] ++
    take (x - y - length [Ancestor (Unite n m)])
      (drop (S y) f)) ++ ReferTo y :: drop (S x) f) =
dsuLeafCount f simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
dsuLeafCount
  ((take y f ++
    take (x - y) [Ancestor (Unite n m)] ++
    take (x - y - 1) (drop (S y) f)) ++
   ReferTo y :: drop (S x) f) = dsuLeafCount f
      assert (stp : take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
take (x - y) [Ancestor (Unite n m)] =
[Ancestor (Unite n m)]
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
dsuLeafCount
  ((take y f ++
    take (x - y) [Ancestor (Unite n m)] ++
    take (x - y - 1) (drop (S y) f)) ++
   ReferTo y :: drop (S x) f) = dsuLeafCount f
      {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
take (x - y) [Ancestor (Unite n m)] =
[Ancestor (Unite n m)] rewrite (ltac:(lia) : x - y = S (x - y - 1)).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
take (S (x - y - 1)) [Ancestor (Unite n m)] =
[Ancestor (Unite n m)] simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
Ancestor (Unite n m) :: take (x - y - 1) [] =
[Ancestor (Unite n m)] rewrite take_nil.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
[Ancestor (Unite n m)] = [Ancestor (Unite n m)] reflexivity. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
dsuLeafCount
  ((take y f ++
    take (x - y) [Ancestor (Unite n m)] ++
    take (x - y - 1) (drop (S y) f)) ++
   ReferTo y :: drop (S x) f) = dsuLeafCount f
      rewrite stp, !take_drop_commute, (ltac:(lia) : S y + (x - y - 1) = x).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
dsuLeafCount
  ((take y f ++
    [Ancestor (Unite n m)] ++ drop (S y) (take x f)) ++
   ReferTo y :: drop (S x) f) = dsuLeafCount f
      rewrite ((ltac:(intros; simpl; easy) : forall a b, a :: b = [a] ++ b) (ReferTo _)).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
dsuLeafCount
  ((take y f ++
    [Ancestor (Unite n m)] ++ drop (S y) (take x f)) ++
   [ReferTo y] ++ drop (S x) f) = dsuLeafCount f
      rewrite (ListDecomposition.listDecomposition f y x jj ltac:(lia) (Ancestor Unit)) at 4.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
dsuLeafCount
  ((take y f ++
    [Ancestor (Unite n m)] ++ drop (S y) (take x f)) ++
   [ReferTo y] ++ drop (S x) f) =
dsuLeafCount
  (take y f ++
   [nth y f (Ancestor Unit)] ++
   drop (S y) (take x f) ++
   [nth x f (Ancestor Unit)] ++ drop (x + 1) f) unfold dsuLeafCount.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end)
        ((take y f ++
          [Ancestor (Unite n m)] ++
          drop (S y) (take x f)) ++
         [ReferTo y] ++ drop (S x) f))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end)
        (take y f ++
         [nth y f (Ancestor Unit)] ++
         drop (S y) (take x f) ++
         [nth x f (Ancestor Unit)] ++ drop (x + 1) f)))
      rewrite !map_app, !list_sum_app.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length (pathCompress dsu (length dsu) a x))
     b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor (Unite n m)]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f)))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo y]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) f)))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth y f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth x f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (x + 1) f)))))) rewrite !pathCompressPreservesLength in yb, zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
yb: Ancestor m =
nth x
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y) (Ancestor Unit)
n: Tree
zb: Ancestor n =
nth y
  (pathCompress
     (pathCompress dsu (length dsu) a x)
     (length dsu) b y) (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
f: list Slot
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor (Unite n m)]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f)))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo y]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) f)))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth y f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth x f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (x + 1) f))))))
      rewrite <- hf in yb, zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor (Unite n m)]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f)))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo y]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) f)))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [nth y f (Ancestor Unit)]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [nth x f (Ancestor Unit)]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (x + 1) f)))))) rewrite <- yb, <- zb.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor (Unite n m)]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f)))) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [ReferTo y]) +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S x) f)))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) [Ancestor n]) +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) [Ancestor m]) +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (x + 1) f))))))
      simpl.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + leafCount m + 0 +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f)))) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) f))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (leafCount m + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (x + 1) f)))))) rewrite (ltac:(lia) : x + 1 = S x).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + leafCount m + 0 +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f)))) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) f))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (leafCount m + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f))))))
      assert (ms : drop (S x) (take (S x) f) = []).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
drop (S x) (take (S x) f) = []
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
ms: drop (S x) (take (S x) f) = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + leafCount m + 0 +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f)))) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) f))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (leafCount m + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f))))))
      {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
drop (S x) (take (S x) f) = [] assert (mt : length (drop (S x) (take (S x) f)) = 0).
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
length (drop (S x) (take (S x) f)) = 0
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
mt: length (drop (S x) (take (S x) f)) = 0
drop (S x) (take (S x) f) = []
        {
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
length (drop (S x) (take (S x) f)) = 0 rewrite drop_length, take_length.
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
S x `min` length f - S x = 0 lia. }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
mt: length (drop (S x) (take (S x) f)) = 0
drop (S x) (take (S x) f) = []
        exact (nil_length_inv _ mt). }
dsu: list Slot
hN: noIllegalIndices dsu
a, b: nat
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
x: nat
hx: x = ancestor dsu (length dsu) a
y: nat
hy: y =
ancestor (pathCompress dsu (length dsu) a x)
  (length (pathCompress dsu (length dsu) a x)) b
h3: x ≠ y
m: Tree
f: list Slot
yb: Ancestor m = nth x f (Ancestor Unit)
n: Tree
zb: Ancestor n = nth y f (Ancestor Unit)
h4: ¬ leafCount n < leafCount m
xL: x < length dsu
yL: y < length dsu
hf: f =
pathCompress (pathCompress dsu (length dsu) a x)
  (length dsu) b y
lp: dsuLeafCount dsu = dsuLeafCount f
cr: length dsu = length f
jj: y < x
stp: take (x - y) [Ancestor (Unite n m)] = [Ancestor (Unite n m)]
ms: drop (S x) (take (S x) f) = []
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + leafCount m + 0 +
    list_sum
      (map
         (λ _0 : Slot,
            match _0 with
            | ReferTo _ => 0
            | Ancestor _1 => leafCount _1
            end) (drop (S y) (take x f)))) +
   list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (drop (S x) f))) =
Z.of_nat
  (list_sum
     (map
        (λ _0 : Slot,
           match _0 with
           | ReferTo _ => 0
           | Ancestor _1 => leafCount _1
           end) (take y f)) +
   (leafCount n + 0 +
    (list_sum
       (map
          (λ _0 : Slot,
             match _0 with
             | ReferTo _ => 0
             | Ancestor _1 => leafCount _1
             end) (drop (S y) (take x f))) +
     (leafCount m + 0 +
      list_sum
        (map
           (λ _0 : Slot,
              match _0 with
              | ReferTo _ => 0
              | Ancestor _1 => leafCount _1
              end) (drop (S x) f))))))
      lia.
Qed.

Lemma unitePreservesNoIllegalIndices (dsu : list Slot) (a b : nat) (h : noIllegalIndices dsu) : noIllegalIndices (unite dsu a b).
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
noIllegalIndices (unite dsu a b)
Proof.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
noIllegalIndices (unite dsu a b)
  unfold unite.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
noIllegalIndices
  (if decide (length dsu ≤ a)
   then dsu
   else
    if decide (length dsu ≤ b)
    then dsu
    else
     if
      decide
        (ancestor dsu (length dsu) a =
         ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
     then
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
     else
      match
        nth (ancestor dsu (length dsu) a)
          (pathCompress
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _0 =>
          match
            nth
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
              (pathCompress
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b)) (Ancestor Unit)
          with
          | ReferTo _ =>
              pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
          | Ancestor _1 =>
              if decide (leafCount _1 < leafCount _0)
              then
               performMerge
                 (pathCompress
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b
                    (ancestor
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a))
                       (length
                          (pathCompress dsu
                             (length dsu) a
                             (ancestor dsu
                                (length dsu) a))) b))
                 _1 _0
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b) (ancestor dsu (length dsu) a)
              else
               performMerge
                 (pathCompress
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b
                    (ancestor
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a))
                       (length
                          (pathCompress dsu
                             (length dsu) a
                             (ancestor dsu
                                (length dsu) a))) b))
                 _0 _1 (ancestor dsu (length dsu) a)
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b)
          end
      end) case_decide as h1.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: length dsu ≤ a
noIllegalIndices dsu
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
noIllegalIndices
  (if decide (length dsu ≤ b)
   then dsu
   else
    if
     decide
       (ancestor dsu (length dsu) a =
        ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    then
     pathCompress
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))) b
       (ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    else
     match
       nth (ancestor dsu (length dsu) a)
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         (Ancestor Unit)
     with
     | ReferTo _ =>
         pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)
     | Ancestor _0 =>
         match
           nth
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) (Ancestor Unit)
         with
         | ReferTo _ =>
             pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
         | Ancestor _1 =>
             if decide (leafCount _1 < leafCount _0)
             then
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _1 _0
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b) (ancestor dsu (length dsu) a)
             else
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _0 _1
                (ancestor dsu (length dsu) a)
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
         end
     end) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: length dsu ≤ a
noIllegalIndices dsu exact h. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
noIllegalIndices
  (if decide (length dsu ≤ b)
   then dsu
   else
    if
     decide
       (ancestor dsu (length dsu) a =
        ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    then
     pathCompress
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))) b
       (ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    else
     match
       nth (ancestor dsu (length dsu) a)
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         (Ancestor Unit)
     with
     | ReferTo _ =>
         pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)
     | Ancestor _0 =>
         match
           nth
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) (Ancestor Unit)
         with
         | ReferTo _ =>
             pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
         | Ancestor _1 =>
             if decide (leafCount _1 < leafCount _0)
             then
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _1 _0
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b) (ancestor dsu (length dsu) a)
             else
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _0 _1
                (ancestor dsu (length dsu) a)
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
         end
     end) case_decide as h2.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: length dsu ≤ b
noIllegalIndices dsu
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
noIllegalIndices
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu (length dsu) a
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _1 _0
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
        end
    end) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: length dsu ≤ b
noIllegalIndices dsu exact h. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
noIllegalIndices
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu (length dsu) a
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _1 _0
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
        end
    end)
  assert (gameover : noIllegalIndices
  (pathCompress
     (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
     (length (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))) b))).
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
noIllegalIndices
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
noIllegalIndices
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu (length dsu) a
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _1 _0
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
        end
    end)
  {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)) rewrite pathCompressPreservesLength at 1.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a)) 
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
    assert (step : noIllegalIndices (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))).
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a)) 
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
    {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) apply pathCompressPreservesNoIllegalIndices.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
noIllegalIndices dsu
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
a < length dsu
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
ancestor dsu (length dsu) a < length dsu {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
noIllegalIndices dsu exact h. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
a < length dsu
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
ancestor dsu (length dsu) a < length dsu {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
a < length dsu lia. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
ancestor dsu (length dsu) a < length dsu {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
ancestor dsu (length dsu) a < length dsu apply ancestorLtLength; (assumption || lia). } }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a)) 
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
    apply pathCompressPreservesNoIllegalIndices.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
b <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))) b <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) exact step. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
b <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))) b <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
b <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) rewrite pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
b < length dsu lia. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))) b <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))) b <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) apply ancestorLtLength; [| rewrite pathCompressPreservesLength; lia].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
step: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) exact step. } }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
noIllegalIndices
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu 
         (length dsu) a 
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                     (length
                      (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                     b)) _1 _0
               (ancestor
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                     (length
                      (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                     b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                  b)
        end
    end)
  case_decide as heq.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
noIllegalIndices
  match
    nth (ancestor dsu (length dsu) a)
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)) exact gameover. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
noIllegalIndices
  match
    nth (ancestor dsu (length dsu) a)
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end
  remember (nth (ancestor dsu (length dsu) a)
      (pathCompress
         (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)))
         b
         (ancestor
            (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)))
            b)) (Ancestor Unit)) as A eqn:hA.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Slot
hA: A =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
noIllegalIndices
  match A with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end
  symmetry in hA.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Slot
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = A
noIllegalIndices
  match A with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end destruct A as [A | A].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: nat
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo A
noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
noIllegalIndices
  match
    nth
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount A)
      then
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 A
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         A _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: nat
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo A
noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)) exact gameover. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
noIllegalIndices
  match
    nth
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount A)
      then
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 A
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         A _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end
  remember (nth
      (ancestor
         (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)))
         b)
      (pathCompress
         (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)))
         b
         (ancestor
            (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)))
            b)) (Ancestor Unit)) as B eqn:hB.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Slot
hB: B =
nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
noIllegalIndices
  match B with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount A)
      then
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 A
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         A _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end
  symmetry in hB.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Slot
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = B
noIllegalIndices
  match B with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount A)
      then
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 A
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         A _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end destruct B as [B | B].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: nat
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo B
noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
noIllegalIndices
  (if decide (leafCount B < leafCount A)
   then
    performMerge
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      B A
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (ancestor dsu (length dsu) a)
   else
    performMerge
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      A B (ancestor dsu (length dsu) a)
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: nat
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo B
noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)) exact gameover. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
noIllegalIndices
  (if decide (leafCount B < leafCount A)
   then
    performMerge
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      B A
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (ancestor dsu (length dsu) a)
   else
    performMerge
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      A B (ancestor dsu (length dsu) a)
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b))
  case_decide as h3; unfold performMerge.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
noIllegalIndices
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
noIllegalIndices
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
  -
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
noIllegalIndices
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) intros g j i.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: nth g
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)))) (Ancestor Unit) = 
ReferTo j
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) destruct (decide (g = ancestor (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))) b)) as [hs | hs].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: nth g
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)))) (Ancestor Unit) = 
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: nth g
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)))) (Ancestor Unit) = 
ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: nth g
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)))) (Ancestor Unit) = 
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) rewrite <- hs, nth_lookup, list_lookup_insert in i.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (Some (ReferTo (ancestor dsu (length dsu) a))) =
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           g)) !! g) = 
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
g <
length
  (<[ancestor dsu (length dsu) a:=
     Ancestor (Unite A B)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b g))
      -
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (Some (ReferTo (ancestor dsu (length dsu) a))) =
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) simpl in i.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: ReferTo (ancestor dsu (length dsu) a) = ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) rewrite !insert_length, !pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: ReferTo (ancestor dsu (length dsu) a) = ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
j < length dsu injection i.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: ReferTo (ancestor dsu (length dsu) a) = ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
ancestor dsu (length dsu) a = j → j < length dsu intro ht.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: ReferTo (ancestor dsu (length dsu) a) = ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
ht: ancestor dsu (length dsu) a = j
j < length dsu rewrite <- ht.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: ReferTo (ancestor dsu (length dsu) a) = ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
ht: ancestor dsu (length dsu) a = j
ancestor dsu (length dsu) a < length dsu apply ancestorLtLength; (assumption || lia).
      -
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           g)) !! g) = 
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
g <
length
  (<[ancestor dsu (length dsu) a:=
     Ancestor (Unite A B)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b g)) rewrite insert_length, hs, pathCompressPreservesLength, pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           g)) !! g) = 
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) 
  (length dsu) b <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) apply ancestorLtLength.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           g)) !! g) = 
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           g)) !! g) = 
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
b <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           g)) !! g) = 
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) apply pathCompressPreservesNoIllegalIndices; try (assumption || lia).
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           g)) !! g) = 
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
ancestor dsu (length dsu) a < length dsu apply ancestorLtLength; try (assumption || lia). }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           g)) !! g) = 
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
b <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
        rewrite pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           g)) !! g) = 
ReferTo j
hs: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
b < length dsu lia. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: nth g
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)))) (Ancestor Unit) = 
ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    rewrite nth_lookup, list_lookup_insert_ne in i; [| lia].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor dsu (length dsu) a:=
     Ancestor (Unite A B)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! g) = ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    destruct (decide (g = ancestor dsu (length dsu) a)) as [ht | ht].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor dsu (length dsu) a:=
     Ancestor (Unite A B)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! g) = ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
ht: g = ancestor dsu (length dsu) a
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor dsu (length dsu) a:=
     Ancestor (Unite A B)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! g) = ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
ht: g ≠ ancestor dsu (length dsu) a
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor dsu (length dsu) a:=
     Ancestor (Unite A B)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! g) = ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
ht: g = ancestor dsu (length dsu) a
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) rewrite ht, list_lookup_insert in i.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (Some (Ancestor (Unite A B))) = 
ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
ht: g = ancestor dsu (length dsu) a
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor dsu (length dsu) a:=
     Ancestor (Unite A B)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! ancestor dsu (length dsu) a) = 
ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
ht: g = ancestor dsu (length dsu) a
ancestor dsu (length dsu) a <
length
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (Some (Ancestor (Unite A B))) = 
ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
ht: g = ancestor dsu (length dsu) a
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) easy. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor dsu (length dsu) a:=
     Ancestor (Unite A B)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! ancestor dsu (length dsu) a) = 
ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
ht: g = ancestor dsu (length dsu) a
ancestor dsu (length dsu) a <
length
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)) rewrite !pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor dsu (length dsu) a:=
     Ancestor (Unite A B)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! ancestor dsu (length dsu) a) = 
ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
ht: g = ancestor dsu (length dsu) a
ancestor dsu (length dsu) a < length dsu apply ancestorLtLength; (assumption || lia). }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor dsu (length dsu) a:=
     Ancestor (Unite A B)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! g) = ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
ht: g ≠ ancestor dsu (length dsu) a
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    rewrite list_lookup_insert_ne, <- nth_lookup in i; [| lia].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: nth g
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
ht: g ≠ ancestor dsu (length dsu) a
j <
length
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a 
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     ReferTo (ancestor dsu (length dsu) a)]>
     (<[ancestor dsu (length dsu) a:=
        Ancestor (Unite A B)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    rewrite !insert_length.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: leafCount B < leafCount A
g, j: nat
i: nth g
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo j
hs: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
ht: g ≠ ancestor dsu (length dsu) a
j <
length
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)) exact (gameover _ _ i).
  -
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
noIllegalIndices
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) intros g j i.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: nth g
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)))) (Ancestor Unit) = 
ReferTo j
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) destruct (decide (g = ancestor dsu (length dsu) a)) as [hs | hs].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: nth g
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)))) (Ancestor Unit) = 
ReferTo j
hs: g = ancestor dsu (length dsu) a
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: nth g
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)))) (Ancestor Unit) = 
ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: nth g
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)))) (Ancestor Unit) = 
ReferTo j
hs: g = ancestor dsu (length dsu) a
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) rewrite <- hs, nth_lookup, list_lookup_insert in i.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (Some
     (ReferTo
        (ancestor
           (pathCompress dsu (length dsu) a g)
           (length
              (pathCompress dsu (length dsu) a g))
           b))) = ReferTo j
hs: g = ancestor dsu (length dsu) a
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo
          (ancestor
             (pathCompress dsu (length dsu) a g)
             (length
                (pathCompress dsu (length dsu) a g))
             b)]>
     (<[ancestor
          (pathCompress dsu (length dsu) a g)
          (length
             (pathCompress dsu (length dsu) a g))
          b:=Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu (length dsu) a g)
           (length
              (pathCompress dsu (length dsu) a g))
           b
           (ancestor
              (pathCompress dsu (length dsu) a g)
              (length
                 (pathCompress dsu 
                    (length dsu) a g)) b))) !! g) =
ReferTo j
hs: g = ancestor dsu (length dsu) a
g <
length
  (<[ancestor (pathCompress dsu (length dsu) a g)
       (length (pathCompress dsu (length dsu) a g)) b:=
     Ancestor (Unite B A)]>
     (pathCompress (pathCompress dsu (length dsu) a g)
        (length (pathCompress dsu (length dsu) a g)) b
        (ancestor (pathCompress dsu (length dsu) a g)
           (length (pathCompress dsu (length dsu) a g))
           b)))
      -
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (Some
     (ReferTo
        (ancestor
           (pathCompress dsu (length dsu) a g)
           (length
              (pathCompress dsu (length dsu) a g))
           b))) = ReferTo j
hs: g = ancestor dsu (length dsu) a
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) simpl in i.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: ReferTo
  (ancestor (pathCompress dsu (length dsu) a g)
     (length (pathCompress dsu (length dsu) a g))
     b) = ReferTo j
hs: g = ancestor dsu (length dsu) a
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) rewrite !insert_length, pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: ReferTo
  (ancestor (pathCompress dsu (length dsu) a g)
     (length (pathCompress dsu (length dsu) a g))
     b) = ReferTo j
hs: g = ancestor dsu (length dsu) a
j <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) injection i.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: ReferTo
  (ancestor (pathCompress dsu (length dsu) a g)
     (length (pathCompress dsu (length dsu) a g))
     b) = ReferTo j
hs: g = ancestor dsu (length dsu) a
ancestor (pathCompress dsu (length dsu) a g)
  (length (pathCompress dsu (length dsu) a g)) b = j
→ j <
  length
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a)) intro ht.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: ReferTo
  (ancestor (pathCompress dsu (length dsu) a g)
     (length (pathCompress dsu (length dsu) a g))
     b) = ReferTo j
hs: g = ancestor dsu (length dsu) a
ht: ancestor (pathCompress dsu (length dsu) a g)
  (length (pathCompress dsu (length dsu) a g)) b =
j
j <
length
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) rewrite <- ht, <- hs.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: ReferTo
  (ancestor (pathCompress dsu (length dsu) a g)
     (length (pathCompress dsu (length dsu) a g))
     b) = ReferTo j
hs: g = ancestor dsu (length dsu) a
ht: ancestor (pathCompress dsu (length dsu) a g)
  (length (pathCompress dsu (length dsu) a g)) b =
j
ancestor (pathCompress dsu (length dsu) a g)
  (length (pathCompress dsu (length dsu) a g)) b <
length (pathCompress dsu (length dsu) a g) apply ancestorLtLength; try rewrite pathCompressPreservesLength; try (assumption || lia).
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: ReferTo
  (ancestor (pathCompress dsu (length dsu) a g)
     (length (pathCompress dsu (length dsu) a g))
     b) = ReferTo j
hs: g = ancestor dsu (length dsu) a
ht: ancestor (pathCompress dsu (length dsu) a g)
  (length (pathCompress dsu (length dsu) a g)) b =
j
noIllegalIndices (pathCompress dsu (length dsu) a g) apply pathCompressPreservesNoIllegalIndices; try (assumption || lia).
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: ReferTo
  (ancestor (pathCompress dsu (length dsu) a g)
     (length (pathCompress dsu (length dsu) a g))
     b) = ReferTo j
hs: g = ancestor dsu (length dsu) a
ht: ancestor (pathCompress dsu (length dsu) a g)
  (length (pathCompress dsu (length dsu) a g)) b =
j
g < length dsu rewrite hs.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: ReferTo
  (ancestor (pathCompress dsu (length dsu) a g)
     (length (pathCompress dsu (length dsu) a g))
     b) = ReferTo j
hs: g = ancestor dsu (length dsu) a
ht: ancestor (pathCompress dsu (length dsu) a g)
  (length (pathCompress dsu (length dsu) a g)) b =
j
ancestor dsu (length dsu) a < length dsu apply ancestorLtLength; (assumption || lia).
      -
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo
          (ancestor
             (pathCompress dsu (length dsu) a g)
             (length
                (pathCompress dsu (length dsu) a g))
             b)]>
     (<[ancestor
          (pathCompress dsu (length dsu) a g)
          (length
             (pathCompress dsu (length dsu) a g))
          b:=Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu (length dsu) a g)
           (length
              (pathCompress dsu (length dsu) a g))
           b
           (ancestor
              (pathCompress dsu (length dsu) a g)
              (length
                 (pathCompress dsu 
                    (length dsu) a g)) b))) !! g) =
ReferTo j
hs: g = ancestor dsu (length dsu) a
g <
length
  (<[ancestor (pathCompress dsu (length dsu) a g)
       (length (pathCompress dsu (length dsu) a g)) b:=
     Ancestor (Unite B A)]>
     (pathCompress (pathCompress dsu (length dsu) a g)
        (length (pathCompress dsu (length dsu) a g)) b
        (ancestor (pathCompress dsu (length dsu) a g)
           (length (pathCompress dsu (length dsu) a g))
           b))) rewrite !insert_length, !pathCompressPreservesLength, hs.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[g:=ReferTo
          (ancestor
             (pathCompress dsu (length dsu) a g)
             (length
                (pathCompress dsu (length dsu) a g))
             b)]>
     (<[ancestor
          (pathCompress dsu (length dsu) a g)
          (length
             (pathCompress dsu (length dsu) a g))
          b:=Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu (length dsu) a g)
           (length
              (pathCompress dsu (length dsu) a g))
           b
           (ancestor
              (pathCompress dsu (length dsu) a g)
              (length
                 (pathCompress dsu 
                    (length dsu) a g)) b))) !! g) =
ReferTo j
hs: g = ancestor dsu (length dsu) a
ancestor dsu (length dsu) a < length dsu apply ancestorLtLength; (assumption || lia). }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: nth g
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)))) (Ancestor Unit) = 
ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    rewrite nth_lookup, list_lookup_insert_ne in i; [| lia].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! g) = ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    destruct (decide (g = ancestor (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))) b)) as [ht | ht].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! g) = ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! g) = ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! g) = ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) rewrite ht, list_lookup_insert in i.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (Some (Ancestor (Unite B A))) = 
ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b) =
ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))) b <
length
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (Some (Ancestor (Unite B A))) = 
ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b)))) easy. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b) =
ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))) b <
length
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)) rewrite !pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b) =
ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) 
  (length dsu) b < length dsu pose proof ancestorLtLength (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)) as t.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b) =
ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
t: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
→ ∀ _0 _1 : nat,
    _1 <
    length
      (pathCompress dsu 
         (length dsu) a
         (ancestor dsu (length dsu) a))
    → ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)) _0 _1 <
      length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) 
  (length dsu) b < length dsu rewrite !pathCompressPreservesLength in t.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b) =
ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
t: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
→ ∀ _0 _1 : nat,
    _1 < length dsu
    → ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)) _0 _1 <
      length dsu
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) 
  (length dsu) b < length dsu apply t; [| lia].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b) =
ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
t: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
→ ∀ _0 _1 : nat,
    _1 < length dsu
    → ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)) _0 _1 <
      length dsu
noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a)) apply pathCompressPreservesNoIllegalIndices; try (assumption || lia).
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b) =
ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))) b
t: noIllegalIndices
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
→ ∀ _0 _1 : nat,
    _1 < length dsu
    → ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)) _0 _1 <
      length dsu
ancestor dsu (length dsu) a < length dsu apply ancestorLtLength; (assumption || lia). }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: default (Ancestor Unit)
  (<[ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b:=
     Ancestor (Unite B A)]>
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
   !! g) = ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    rewrite list_lookup_insert_ne, <- nth_lookup in i; [| lia].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: nth g
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
j <
length
  (<[ancestor dsu (length dsu) a:=
     ReferTo
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b)]>
     (<[ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b:=
        Ancestor (Unite B A)]>
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))))
    rewrite !insert_length.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: ¬ length dsu ≤ a
h2: ¬ length dsu ≤ b
gameover: noIllegalIndices
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
heq: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
A: Tree
hA: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor A
B: Tree
hB: nth
  (ancestor
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor B
h3: ¬ leafCount B < leafCount A
g, j: nat
i: nth g
  (pathCompress
     (pathCompress dsu 
        (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo j
hs: g ≠ ancestor dsu (length dsu) a
ht: g
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
j <
length
  (pathCompress
     (pathCompress dsu 
        (length dsu) a 
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)) exact (gameover _ _ i).
Qed.

Lemma performMergePreservesWithoutCycles (dsu : list Slot) (a b : nat) (ha : a < length dsu) (hb : b < length dsu) (hD : a <> b) t1 t2 (h : noIllegalIndices dsu) (h1 : withoutCyclesN dsu (length dsu)) t3 (h3 : nth a dsu (Ancestor Unit) = Ancestor t3) t4 (h4 : nth b dsu (Ancestor Unit) = Ancestor t4) : withoutCyclesN (performMerge dsu t1 t2 a b) (length dsu).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
withoutCyclesN (performMerge dsu t1 t2 a b)
  (length dsu)
Proof.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
withoutCyclesN (performMerge dsu t1 t2 a b)
  (length dsu)
  unfold performMerge.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
withoutCyclesN
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu)) (length dsu)
  intros i j.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  assert (als : noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
noIllegalIndices
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
noIllegalIndices
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu)) intros yy rr hh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: nth yy
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo rr
rr <
length
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
    rewrite !insert_length.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: nth yy
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo rr
rr < length dsu
    destruct (Nat.eq_dec yy a) as [ww | ww].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: nth yy
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo rr
ww: yy = a
rr < length dsu
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: nth yy
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo rr
ww: yy ≠ a
rr < length dsu
    -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: nth yy
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo rr
ww: yy = a
rr < length dsu rewrite ww in hh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: nth a
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo rr
ww: yy = a
rr < length dsu
      rewrite nth_lookup, list_lookup_insert in hh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: default (Ancestor Unit) (Some (ReferTo b)) =
ReferTo rr
ww: yy = a
rr < length dsu
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: default (Ancestor Unit)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu) !! a) =
ReferTo rr
ww: yy = a
a < length (<[b:=Ancestor (Unite t2 t1)]> dsu)
      +
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: default (Ancestor Unit) (Some (ReferTo b)) =
ReferTo rr
ww: yy = a
rr < length dsu simpl in hh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: ReferTo b = ReferTo rr
ww: yy = a
rr < length dsu
        injection hh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: ReferTo b = ReferTo rr
ww: yy = a
b = rr → rr < length dsu
        intro als.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: ReferTo b = ReferTo rr
ww: yy = a
als: b = rr
rr < length dsu
        subst b.
dsu: list Slot
a: nat
ha: a < length dsu
rr: nat
hh: ReferTo rr = ReferTo rr
hD: a ≠ rr
hb: rr < length dsu
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth rr dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy: nat
ww: yy = a
rr < length dsu
        pose proof ancestorLtLength dsu ltac:(assumption) (length dsu) i ltac:(lia).
dsu: list Slot
a: nat
ha: a < length dsu
rr: nat
hh: ReferTo rr = ReferTo rr
hD: a ≠ rr
hb: rr < length dsu
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth rr dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy: nat
ww: yy = a
_0: ancestor dsu (length dsu) i < length dsu
rr < length dsu lia.
      +
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: default (Ancestor Unit)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu) !! a) =
ReferTo rr
ww: yy = a
a < length (<[b:=Ancestor (Unite t2 t1)]> dsu) rewrite insert_length.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: default (Ancestor Unit)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu) !! a) =
ReferTo rr
ww: yy = a
a < length dsu lia.
    -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: nth yy
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo rr
ww: yy ≠ a
rr < length dsu rewrite nth_lookup in hh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: default (Ancestor Unit)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu) !! yy) =
ReferTo rr
ww: yy ≠ a
rr < length dsu
      destruct (Nat.eq_dec yy b) as [ii | ii].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: default (Ancestor Unit)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu) !! yy) =
ReferTo rr
ww: yy ≠ a
ii: yy = b
rr < length dsu
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: default (Ancestor Unit)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu) !! yy) =
ReferTo rr
ww: yy ≠ a
ii: yy ≠ b
rr < length dsu
      +
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: default (Ancestor Unit)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu) !! yy) =
ReferTo rr
ww: yy ≠ a
ii: yy = b
rr < length dsu subst yy.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: default (Ancestor Unit)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu) !! b) =
ReferTo rr
rr < length dsu
        rewrite (list_lookup_insert_ne _ _ _ _ (ltac:(lia) : a ≠ b)), list_lookup_insert in hh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: default (Ancestor Unit)
  (Some (Ancestor (Unite t2 t1))) = ReferTo rr
rr < length dsu
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: default (Ancestor Unit)
  (<[b:=Ancestor (Unite t2 t1)]> dsu !! b) =
ReferTo rr
b < length dsu
        *
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: default (Ancestor Unit)
  (Some (Ancestor (Unite t2 t1))) = ReferTo rr
rr < length dsu simpl in hh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: Ancestor (Unite t2 t1) = ReferTo rr
rr < length dsu easy.
        *
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: default (Ancestor Unit)
  (<[b:=Ancestor (Unite t2 t1)]> dsu !! b) =
ReferTo rr
b < length dsu rewrite list_lookup_insert in hh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: default (Ancestor Unit)
  (Some (Ancestor (Unite t2 t1))) = ReferTo rr
b < length dsu
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: default (Ancestor Unit)
  (<[b:=Ancestor (Unite t2 t1)]> dsu !! b) =
ReferTo rr
b < length dsu
          {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: default (Ancestor Unit)
  (Some (Ancestor (Unite t2 t1))) = ReferTo rr
b < length dsu simpl in hh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: Ancestor (Unite t2 t1) = ReferTo rr
b < length dsu easy. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: default (Ancestor Unit)
  (<[b:=Ancestor (Unite t2 t1)]> dsu !! b) =
ReferTo rr
b < length dsu
          {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
rr: nat
ww: b ≠ a
hh: default (Ancestor Unit)
  (<[b:=Ancestor (Unite t2 t1)]> dsu !! b) =
ReferTo rr
b < length dsu lia. }
      +
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: default (Ancestor Unit)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu) !! yy) =
ReferTo rr
ww: yy ≠ a
ii: yy ≠ b
rr < length dsu rewrite (list_lookup_insert_ne _ _ _ _ (Nat.neq_sym _ _ ww)), (list_lookup_insert_ne _ _ _ _ (Nat.neq_sym _ _ ii)), <- nth_lookup in hh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
yy, rr: nat
hh: nth yy dsu (Ancestor Unit) = ReferTo rr
ww: yy ≠ a
ii: yy ≠ b
rr < length dsu
        exact (h _ _ hh). }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  assert (hL : 2 <= length dsu).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
2 ≤ length dsu
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
2 ≤ length dsu destruct dsu as [| head [| head' tail]]; simpl in *; lia. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  destruct (Nat.eq_dec (ancestor dsu (length dsu) i) a) as [hs | hs].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end assert (ya : validChainToAncestor (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu)) (ancestorChain dsu (length dsu) i ++ [b])).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i ++ [b])
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
ya: validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i ++ [b])
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
    {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i ++ [b]) pose proof validChainAncestorChain dsu (length dsu) i ltac:(assumption) ltac:(assumption) as [g1 [g2 g3]].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i ++ [b])
      repeat split.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ancestorChain dsu (length dsu) i ++ [b] ≠ []
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth 0 (ancestorChain dsu (length dsu) i ++ [b]) 0 <
length
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i ++ [b])
  → nth
      (nth _0
         (ancestorChain dsu (length dsu) i ++ [b]) 0)
      (<[a:=ReferTo b]>
         (<[b:=Ancestor (Unite t2 t1)]> dsu))
      (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i ++ [b]) 0)
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∃ _0 : Tree,
  nth
    (nth
       (length
          (ancestorChain dsu (length dsu) i ++ [b]) -
        1) (ancestorChain dsu (length dsu) i ++ [b]) 0)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit) = Ancestor _0
      -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ancestorChain dsu (length dsu) i ++ [b] ≠ [] intro ga.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ga: ancestorChain dsu (length dsu) i ++ [b] = []
False
        assert (ayw : length (ancestorChain dsu (length dsu) i ++ [b]) = length ([] : list nat)).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ga: ancestorChain dsu (length dsu) i ++ [b] = []
length (ancestorChain dsu (length dsu) i ++ [b]) =
length []
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ga: ancestorChain dsu (length dsu) i ++ [b] = []
ayw: length (ancestorChain dsu (length dsu) i ++ [b]) = length ([] : list nat)
False
        {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ga: ancestorChain dsu (length dsu) i ++ [b] = []
length (ancestorChain dsu (length dsu) i ++ [b]) =
length [] rewrite ga.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ga: ancestorChain dsu (length dsu) i ++ [b] = []
length [] = length [] reflexivity. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ga: ancestorChain dsu (length dsu) i ++ [b] = []
ayw: length (ancestorChain dsu (length dsu) i ++ [b]) = length ([] : list nat)
False
        rewrite app_length in ayw.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ga: ancestorChain dsu (length dsu) i ++ [b] = []
ayw: length (ancestorChain dsu (length dsu) i) + length [b] = length []
False simpl in ayw.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ga: ancestorChain dsu (length dsu) i ++ [b] = []
ayw: length (ancestorChain dsu (length dsu) i) + 1 = 0
False lia.
      -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth 0 (ancestorChain dsu (length dsu) i ++ [b]) 0 <
length
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu)) rewrite !insert_length, nth_lookup, lookup_app_l.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
default 0 (ancestorChain dsu (length dsu) i !! 0) <
length dsu
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
0 < length (ancestorChain dsu (length dsu) i)
        +
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
default 0 (ancestorChain dsu (length dsu) i !! 0) <
length dsu rewrite <- nth_lookup.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu exact g2.
        +
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
0 < length (ancestorChain dsu (length dsu) i) pose proof (ltac:(lia) : 0 < length (ancestorChain dsu (length dsu) i) \/ length (ancestorChain dsu (length dsu) i) = 0) as [ds | ds].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ds: 0 < length (ancestorChain dsu (length dsu) i)
0 < length (ancestorChain dsu (length dsu) i)
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ds: length (ancestorChain dsu (length dsu) i) = 0
0 < length (ancestorChain dsu (length dsu) i) {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ds: 0 < length (ancestorChain dsu (length dsu) i)
0 < length (ancestorChain dsu (length dsu) i) exact ds. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ds: length (ancestorChain dsu (length dsu) i) = 0
0 < length (ancestorChain dsu (length dsu) i)
          pose proof nil_length_inv _ ds.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ds: length (ancestorChain dsu (length dsu) i) = 0
_0: ancestorChain dsu (length dsu) i = []
0 < length (ancestorChain dsu (length dsu) i) tauto.
      -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i ++ [b])
  → nth
      (nth _0
         (ancestorChain dsu (length dsu) i ++ [b]) 0)
      (<[a:=ReferTo b]>
         (<[b:=Ancestor (Unite t2 t1)]> dsu))
      (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i ++ [b]) 0) intros ja k.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i ++ [b])
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja)
     (ancestorChain dsu (length dsu) i ++ [b]) 0) rewrite app_length in k.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) +
length [b]
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja)
     (ancestorChain dsu (length dsu) i ++ [b]) 0) simpl in k.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja)
     (ancestorChain dsu (length dsu) i ++ [b]) 0)
        pose proof (ltac:(lia) : S ja < length (ancestorChain dsu (length dsu) i) \/ S ja = length (ancestorChain dsu (length dsu) i)) as [u | u].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja)
     (ancestorChain dsu (length dsu) i ++ [b]) 0)
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja)
     (ancestorChain dsu (length dsu) i ++ [b]) 0)
        +
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja)
     (ancestorChain dsu (length dsu) i ++ [b]) 0) rewrite !nth_lookup, lookup_app_l, lookup_app_l, <- !nth_lookup; [| lia | lia].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i) 0)
          pose proof g3 ja ltac:(lia) as g4.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i) 0)
          assert (n1 : nth ja (ancestorChain dsu (length dsu) i) 0 <> a).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i) 0)
          {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a intro n1.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 = a
False rewrite n1 in g4.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth a dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 = a
False rewrite g4 in h3.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
i, ja: nat
h3: ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth a dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 = a
False easy. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i) 0)
          assert (n2 : nth ja (ancestorChain dsu (length dsu) i) 0 <> b).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i) 0)
          {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b intro n2.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 = b
False rewrite n2 in g4.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth b dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 = b
False rewrite g4 in h4.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
i, ja: nat
h4: ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0) = Ancestor t4
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth b dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 = b
False easy. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i) 0)
          rewrite !nth_lookup, !list_lookup_insert_ne, <- !nth_lookup.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
nth (nth ja (ancestorChain dsu (length dsu) i) 0) dsu
  (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i) 0)
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
b ≠ default 0 (ancestorChain dsu (length dsu) i !! ja)
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
a ≠ default 0 (ancestorChain dsu (length dsu) i !! ja) {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
nth (nth ja (ancestorChain dsu (length dsu) i) 0) dsu
  (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i) 0) assumption. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
b ≠ default 0 (ancestorChain dsu (length dsu) i !! ja)
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
a ≠ default 0 (ancestorChain dsu (length dsu) i !! ja)
          *
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
b ≠ default 0 (ancestorChain dsu (length dsu) i !! ja) rewrite <- nth_lookup.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
b ≠ nth ja (ancestorChain dsu (length dsu) i) 0 lia.
          *
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
a ≠ default 0 (ancestorChain dsu (length dsu) i !! ja) rewrite <- nth_lookup.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja < length (ancestorChain dsu (length dsu) i)
g4: nth (nth ja (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S ja) (ancestorChain dsu (length dsu) i)
     0)
n1: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth ja (ancestorChain dsu (length dsu) i) 0 ≠ b
a ≠ nth ja (ancestorChain dsu (length dsu) i) 0 lia.
        +
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja)
     (ancestorChain dsu (length dsu) i ++ [b]) 0) assert (stp : nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 =
b
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja)
     (ancestorChain dsu (length dsu) i ++ [b]) 0)
          {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 =
b rewrite nth_lookup, lookup_app_r.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
default 0
  ([b]
   !! (S ja -
       length (ancestorChain dsu (length dsu) i))) = b
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
length (ancestorChain dsu (length dsu) i) ≤ S ja {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
default 0
  ([b]
   !! (S ja -
       length (ancestorChain dsu (length dsu) i))) = b rewrite <- u, (ltac:(lia) : S ja - S ja = 0).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
default 0 ([b] !! 0) = b easy. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
length (ancestorChain dsu (length dsu) i) ≤ S ja lia. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S ja)
     (ancestorChain dsu (length dsu) i ++ [b]) 0)
          rewrite stp.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo b
          pose proof ancestorEqLastAncestorChain dsu (length dsu) i as gh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 =
ancestor dsu (length dsu) i
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo b
          rewrite hs in gh.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo b
          assert (spt : nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0 = a).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0 = a
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
spt: nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0 = a
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo b
          {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0 = a rewrite nth_lookup, lookup_app_l, (ltac:(lia) : ja = length (ancestorChain dsu (length dsu) i) - 1), <- nth_lookup.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
nth (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
ja < length (ancestorChain dsu (length dsu) i) {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
nth (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a exact gh. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
ja < length (ancestorChain dsu (length dsu) i) lia. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
spt: nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0 = a
nth
  (nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = ReferTo b
          rewrite spt, nth_lookup, list_lookup_insert.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
spt: nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0 = a
default (Ancestor Unit) (Some (ReferTo b)) = ReferTo b
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
spt: nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0 = a
a < length (<[b:=Ancestor (Unite t2 t1)]> dsu) {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
spt: nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0 = a
default (Ancestor Unit) (Some (ReferTo b)) = ReferTo b easy. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
spt: nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0 = a
a < length (<[b:=Ancestor (Unite t2 t1)]> dsu) rewrite insert_length.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ja: nat
k: S ja <
length (ancestorChain dsu (length dsu) i) + 1
u: S ja = length (ancestorChain dsu (length dsu) i)
stp: nth (S ja) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
gh: nth
  (length (ancestorChain dsu (length dsu) i) - 1)
  (ancestorChain dsu (length dsu) i) 0 = a
spt: nth ja (ancestorChain dsu (length dsu) i ++ [b]) 0 = a
a < length dsu exact ha.
      -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∃ _0 : Tree,
  nth
    (nth
       (length
          (ancestorChain dsu (length dsu) i ++ [b]) -
        1) (ancestorChain dsu (length dsu) i ++ [b]) 0)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit) = Ancestor _0 exists (Unite t2 t1).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth
  (nth
     (length (ancestorChain dsu (length dsu) i ++ [b]) -
      1) (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = Ancestor (Unite t2 t1)
        assert (ajk : nth (length (ancestorChain dsu (length dsu) i ++ [b]) - 1) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth
  (length (ancestorChain dsu (length dsu) i ++ [b]) -
   1) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ajk: nth (length (ancestorChain dsu (length dsu) i ++ [b]) - 1)
(ancestorChain dsu (length dsu) i ++ [b]) 0 = b
nth
  (nth
     (length (ancestorChain dsu (length dsu) i ++ [b]) -
      1) (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = Ancestor (Unite t2 t1)
        {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth
  (length (ancestorChain dsu (length dsu) i ++ [b]) -
   1) (ancestorChain dsu (length dsu) i ++ [b]) 0 = b rewrite app_length.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth
  (length (ancestorChain dsu (length dsu) i) +
   length [b] - 1)
  (ancestorChain dsu (length dsu) i ++ [b]) 0 = b simpl.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth
  (length (ancestorChain dsu (length dsu) i) + 1 - 1)
  (ancestorChain dsu (length dsu) i ++ [b]) 0 = b rewrite Nat.add_sub, nth_lookup, lookup_app_r, Nat.sub_diag.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
default 0 ([b] !! 0) = b
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
length (ancestorChain dsu (length dsu) i)
≤ length (ancestorChain dsu (length dsu) i) {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
default 0 ([b] !! 0) = b easy. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
length (ancestorChain dsu (length dsu) i)
≤ length (ancestorChain dsu (length dsu) i) lia. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ajk: nth (length (ancestorChain dsu (length dsu) i ++ [b]) - 1)
(ancestorChain dsu (length dsu) i ++ [b]) 0 = b
nth
  (nth
     (length (ancestorChain dsu (length dsu) i ++ [b]) -
      1) (ancestorChain dsu (length dsu) i ++ [b]) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = Ancestor (Unite t2 t1)
        rewrite ajk, nth_lookup, list_lookup_insert_ne, list_lookup_insert; (easy || lia). }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
ya: validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i ++ [b])
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
    pose proof validChainAncestorLength (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu)) (ancestorChain dsu (length dsu) i ++ [b]) als ya i ltac:(rewrite nth_lookup, lookup_app_l; [destruct (length dsu) as [| g]; [simpl; easy |]; simpl; destruct (nth i dsu (Ancestor Unit)) as [jk | jk]; easy | apply zeroLtLengthAncestorChain]) as ua.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
ya: validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i ++ [b])
ua: ancestorChain dsu (length dsu) i ++ [b] =
ancestorChain
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length
     (<[a:=ReferTo b]>
        (<[b:=Ancestor (Unite t2 t1)]> dsu))) i
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite !insert_length in ua.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
ya: validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i ++ [b])
ua: ancestorChain dsu (length dsu) i ++ [b] =
ancestorChain
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length dsu) i
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
    pose proof ancestorEqLastAncestorChain (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu)) (length dsu) i as ra.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
ya: validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i ++ [b])
ua: ancestorChain dsu (length dsu) i ++ [b] =
ancestorChain
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length dsu) i
ra: nth
  (length
     (ancestorChain
        (<[a:=ReferTo b]>
           (<[b:=Ancestor (Unite t2 t1)]> dsu))
        (length dsu) i) - 1)
  (ancestorChain
     (<[a:=ReferTo b]>
        (<[b:=Ancestor (Unite t2 t1)]> dsu))
     (length dsu) i) 0 =
ancestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length dsu) i
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
    rewrite <- !ua, nth_lookup, lookup_app_r, app_length, (ltac:(easy) : length [b] = 1), Nat.add_sub, Nat.sub_diag in ra; [| rewrite app_length; simpl; lia].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
ya: validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i ++ [b])
ua: ancestorChain dsu (length dsu) i ++ [b] =
ancestorChain
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length dsu) i
ra: default 0 ([b] !! 0) =
ancestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length dsu) i
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end symmetry in ra.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
ya: validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i ++ [b])
ua: ancestorChain dsu (length dsu) i ++ [b] =
ancestorChain
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length dsu) i
ra: ancestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length dsu) i = default 0 ([b] !! 0)
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite (ltac:(easy) : default 0 ([b] !! 0) = b) in ra.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i = a
ya: validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i ++ [b])
ua: ancestorChain dsu (length dsu) i ++ [b] =
ancestorChain
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length dsu) i
ra: ancestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length dsu) i = b
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end rewrite !insert_length, ra, nth_lookup, list_lookup_insert_ne, list_lookup_insert; (easy || lia). }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  destruct (Nat.eq_dec (ancestor dsu (length dsu) i) b) as [ht | ht].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end assert (ter: validChainToAncestor (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu)) (ancestorChain dsu (length dsu) i)).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i)
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
ter: validChainToAncestor (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
(ancestorChain dsu (length dsu) i)
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
    {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i) pose proof validChainAncestorChain dsu (length dsu) i ltac:(assumption) ltac:(assumption) as [g1 [g2 g3]].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i)
      repeat split.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ancestorChain dsu (length dsu) i ≠ []
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth 0 (ancestorChain dsu (length dsu) i) 0 <
length
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) i)
  → nth (nth _0 (ancestorChain dsu (length dsu) i) 0)
      (<[a:=ReferTo b]>
         (<[b:=Ancestor (Unite t2 t1)]> dsu))
      (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu (length dsu) i) 0)
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) i) - 1)
       (ancestorChain dsu (length dsu) i) 0)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit) = Ancestor _0
      -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ancestorChain dsu (length dsu) i ≠ [] destruct (length dsu) as [| g].
dsu: list Slot
a, b: nat
ha: a < 0
hb: b < 0
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu 0
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < 0
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ 0
hs: ancestor dsu 0 i ≠ a
ht: ancestor dsu 0 i = b
g1: ancestorChain dsu 0 i ≠ []
g2: nth 0 (ancestorChain dsu 0 i) 0 < 0
g3: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu 0 i)
  → nth (nth _0 (ancestorChain dsu 0 i) 0) dsu
      (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu 0 i) 0)
ancestorChain dsu 0 i ≠ []
dsu: list Slot
a, b, g: nat
ha: a < S g
hb: b < S g
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (S g)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < S g
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ S g
hs: ancestor dsu (S g) i ≠ a
ht: ancestor dsu (S g) i = b
g1: ancestorChain dsu (S g) i ≠ []
g2: nth 0 (ancestorChain dsu (S g) i) 0 < S g
g3: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (S g) i)
  → nth (nth _0 (ancestorChain dsu (S g) i) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu (S g) i) 0)
ancestorChain dsu (S g) i ≠ [] {
dsu: list Slot
a, b: nat
ha: a < 0
hb: b < 0
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu 0
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < 0
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ 0
hs: ancestor dsu 0 i ≠ a
ht: ancestor dsu 0 i = b
g1: ancestorChain dsu 0 i ≠ []
g2: nth 0 (ancestorChain dsu 0 i) 0 < 0
g3: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu 0 i)
  → nth (nth _0 (ancestorChain dsu 0 i) 0) dsu
      (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu 0 i) 0)
ancestorChain dsu 0 i ≠ [] simpl.
dsu: list Slot
a, b: nat
ha: a < 0
hb: b < 0
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu 0
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < 0
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ 0
hs: ancestor dsu 0 i ≠ a
ht: ancestor dsu 0 i = b
g1: ancestorChain dsu 0 i ≠ []
g2: nth 0 (ancestorChain dsu 0 i) 0 < 0
g3: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu 0 i)
  → nth (nth _0 (ancestorChain dsu 0 i) 0) dsu
      (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu 0 i) 0)
[i] ≠ [] easy. }
dsu: list Slot
a, b, g: nat
ha: a < S g
hb: b < S g
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (S g)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < S g
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ S g
hs: ancestor dsu (S g) i ≠ a
ht: ancestor dsu (S g) i = b
g1: ancestorChain dsu (S g) i ≠ []
g2: nth 0 (ancestorChain dsu (S g) i) 0 < S g
g3: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (S g) i)
  → nth (nth _0 (ancestorChain dsu (S g) i) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu (S g) i) 0)
ancestorChain dsu (S g) i ≠ [] simpl.
dsu: list Slot
a, b, g: nat
ha: a < S g
hb: b < S g
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (S g)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < S g
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ S g
hs: ancestor dsu (S g) i ≠ a
ht: ancestor dsu (S g) i = b
g1: ancestorChain dsu (S g) i ≠ []
g2: nth 0 (ancestorChain dsu (S g) i) 0 < S g
g3: ∀ _0 : nat,
  S _0 < length (ancestorChain dsu (S g) i)
  → nth (nth _0 (ancestorChain dsu (S g) i) 0)
      dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu (S g) i) 0)
match nth i dsu (Ancestor Unit) with
| ReferTo _0 => i :: ancestorChain dsu g _0
| Ancestor _ => [i]
end ≠ [] destruct (nth i dsu (Ancestor Unit)) as [jk | jk]; easy.
      -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth 0 (ancestorChain dsu (length dsu) i) 0 <
length
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu)) rewrite !insert_length.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu assumption.
      -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) i)
  → nth (nth _0 (ancestorChain dsu (length dsu) i) 0)
      (<[a:=ReferTo b]>
         (<[b:=Ancestor (Unite t2 t1)]> dsu))
      (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu (length dsu) i) 0) intros y u.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
        pose proof g3 y u as su.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
        assert (n1 : nth y (ancestorChain dsu (length dsu) i) 0 <> a).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
        {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
nth y (ancestorChain dsu (length dsu) i) 0 ≠ a intro w.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
w: nth y (ancestorChain dsu (length dsu) i) 0 = a
False rewrite w, h3 in su.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: Ancestor t3 =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
w: nth y (ancestorChain dsu (length dsu) i) 0 = a
False easy. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
        assert (n2 : nth y (ancestorChain dsu (length dsu) i) 0 <> b).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
nth y (ancestorChain dsu (length dsu) i) 0 ≠ b
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth y (ancestorChain dsu (length dsu) i) 0 ≠ b
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
        {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
nth y (ancestorChain dsu (length dsu) i) 0 ≠ b intro w.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
w: nth y (ancestorChain dsu (length dsu) i) 0 = b
False rewrite w, h4 in su.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: Ancestor t4 =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
w: nth y (ancestorChain dsu (length dsu) i) 0 = b
False easy. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
su: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth y (ancestorChain dsu (length dsu) i) 0 ≠ b
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
        rewrite !nth_lookup, !list_lookup_insert_ne, <- !nth_lookup, su; [easy | |]; rewrite <- nth_lookup; lia.
      -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) i) - 1)
       (ancestorChain dsu (length dsu) i) 0)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit) = Ancestor _0 exists (Unite t2 t1).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth
  (nth (length (ancestorChain dsu (length dsu) i) - 1)
     (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) = Ancestor (Unite t2 t1)
        rewrite ancestorEqLastAncestorChain, ht, !nth_lookup, list_lookup_insert_ne, list_lookup_insert; [easy | |]; lia. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
ter: validChainToAncestor (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
(ancestorChain dsu (length dsu) i)
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
    pose proof validChainAncestorLength (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu)) (ancestorChain dsu (length dsu) i) als ter i ltac:(destruct (length dsu); simpl; destruct (nth i dsu (Ancestor Unit)); simpl; try lia) as nn.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i = b
ter: validChainToAncestor (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
(ancestorChain dsu (length dsu) i)
nn: ancestorChain dsu (length dsu) i =
ancestorChain
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length
     (<[a:=ReferTo b]>
        (<[b:=Ancestor (Unite t2 t1)]> dsu))) i
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
    rewrite <- ancestorEqLastAncestorChain, <- nn, ancestorEqLastAncestorChain, ht, !nth_lookup, list_lookup_insert_ne, list_lookup_insert; (easy || lia). }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  assert (ter : validChainToAncestor (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu)) (ancestorChain dsu (length dsu) i)).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i)
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
ter: validChainToAncestor (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
(ancestorChain dsu (length dsu) i)
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i) pose proof validChainAncestorChain dsu (length dsu) i ltac:(assumption) ltac:(assumption) as [g1 [g2 g3]].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
validChainToAncestor
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (ancestorChain dsu (length dsu) i)
    repeat split.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ancestorChain dsu (length dsu) i ≠ []
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth 0 (ancestorChain dsu (length dsu) i) 0 <
length
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) i)
  → nth (nth _0 (ancestorChain dsu (length dsu) i) 0)
      (<[a:=ReferTo b]>
         (<[b:=Ancestor (Unite t2 t1)]> dsu))
      (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu (length dsu) i) 0)
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) i) - 1)
       (ancestorChain dsu (length dsu) i) 0)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit) = Ancestor _0
    -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ancestorChain dsu (length dsu) i ≠ [] destruct (length dsu); try easy; simpl; destruct (nth i _ _); easy.
    -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth 0 (ancestorChain dsu (length dsu) i) 0 <
length
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu)) rewrite !insert_length.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu assumption.
    -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∀ _0 : nat,
  S _0 < length (ancestorChain dsu (length dsu) i)
  → nth (nth _0 (ancestorChain dsu (length dsu) i) 0)
      (<[a:=ReferTo b]>
         (<[b:=Ancestor (Unite t2 t1)]> dsu))
      (Ancestor Unit) =
    ReferTo
      (nth (S _0) (ancestorChain dsu (length dsu) i) 0) intros y u.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
      pose proof g3 y u as us.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
      assert (n1 : nth y (ancestorChain dsu (length dsu) i) 0 <> a).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
      {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
nth y (ancestorChain dsu (length dsu) i) 0 ≠ a intro g.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
g: nth y (ancestorChain dsu (length dsu) i) 0 = a
False rewrite g, h3 in us.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: Ancestor t3 =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
g: nth y (ancestorChain dsu (length dsu) i) 0 = a
False easy. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
      assert (n2 : nth y (ancestorChain dsu (length dsu) i) 0 <> b).
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
nth y (ancestorChain dsu (length dsu) i) 0 ≠ b
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth y (ancestorChain dsu (length dsu) i) 0 ≠ b
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
      {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
nth y (ancestorChain dsu (length dsu) i) 0 ≠ b intro g.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
g: nth y (ancestorChain dsu (length dsu) i) 0 = b
False rewrite g, h4 in us.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: Ancestor t4 =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
g: nth y (ancestorChain dsu (length dsu) i) 0 = b
False easy. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
y: nat
u: S y < length (ancestorChain dsu (length dsu) i)
us: nth (nth y (ancestorChain dsu (length dsu) i) 0)
  dsu (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
n1: nth y (ancestorChain dsu (length dsu) i) 0 ≠ a
n2: nth y (ancestorChain dsu (length dsu) i) 0 ≠ b
nth (nth y (ancestorChain dsu (length dsu) i) 0)
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (Ancestor Unit) =
ReferTo
  (nth (S y) (ancestorChain dsu (length dsu) i) 0)
      rewrite nth_lookup, !list_lookup_insert_ne; [rewrite <- nth_lookup; assumption | |]; lia.
    -
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∃ _0 : Tree,
  nth
    (nth
       (length (ancestorChain dsu (length dsu) i) - 1)
       (ancestorChain dsu (length dsu) i) 0)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit) = Ancestor _0 rewrite ancestorEqLastAncestorChain, nth_lookup, !list_lookup_insert_ne; [| lia | lia].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∃ _0 : Tree,
  default (Ancestor Unit)
    (dsu !! ancestor dsu (length dsu) i) = Ancestor _0 rewrite <- nth_lookup.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
∃ _0 : Tree,
  nth (ancestor dsu (length dsu) i) dsu
    (Ancestor Unit) = Ancestor _0
      pose proof h1 i j as ji.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
ji: match
  nth (ancestor dsu (length dsu) i) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
∃ _0 : Tree,
  nth (ancestor dsu (length dsu) i) dsu
    (Ancestor Unit) = Ancestor _0
      destruct (nth (ancestor dsu (length dsu) i) dsu (Ancestor Unit)) as [nn | nn].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nn: nat
ji: false
∃ _0 : Tree, ReferTo nn = Ancestor _0
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nn: Tree
ji: true
∃ _0 : Tree, Ancestor nn = Ancestor _0 {
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nn: nat
ji: false
∃ _0 : Tree, ReferTo nn = Ancestor _0 easy. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nn: Tree
ji: true
∃ _0 : Tree, Ancestor nn = Ancestor _0 exists nn.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
g1: ancestorChain dsu (length dsu) i ≠ []
g2: nth 0 (ancestorChain dsu (length dsu) i) 0 <
length dsu
g3: ∀ _0 : nat,
  S _0 <
  length (ancestorChain dsu (length dsu) i)
  → nth
      (nth _0 (ancestorChain dsu (length dsu) i)
         0) dsu (Ancestor Unit) =
    ReferTo
      (nth (S _0)
         (ancestorChain dsu (length dsu) i) 0)
nn: Tree
ji: true
Ancestor nn = Ancestor nn reflexivity. }
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
ter: validChainToAncestor (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
(ancestorChain dsu (length dsu) i)
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  pose proof validChainAncestorLength (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu)) (ancestorChain dsu (length dsu) i) als ter i ltac:(destruct (length dsu); simpl; destruct (nth i dsu (Ancestor Unit)); simpl; try lia) as nn.
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
ter: validChainToAncestor (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
(ancestorChain dsu (length dsu) i)
nn: ancestorChain dsu (length dsu) i =
ancestorChain
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length
     (<[a:=ReferTo b]>
        (<[b:=Ancestor (Unite t2 t1)]> dsu))) i
match
  nth
    (ancestor
       (<[a:=ReferTo b]>
          (<[b:=Ancestor (Unite t2 t1)]> dsu))
       (length
          (<[a:=ReferTo b]>
             (<[b:=Ancestor (Unite t2 t1)]> dsu))) i)
    (<[a:=ReferTo b]>
       (<[b:=Ancestor (Unite t2 t1)]> dsu))
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end
  rewrite <- ancestorEqLastAncestorChain, <- nn, ancestorEqLastAncestorChain, nth_lookup, !list_lookup_insert_ne, <- nth_lookup; [| lia | lia].
dsu: list Slot
a, b: nat
ha: a < length dsu
hb: b < length dsu
hD: a ≠ b
t1, t2: Tree
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
t3: Tree
h3: nth a dsu (Ancestor Unit) = Ancestor t3
t4: Tree
h4: nth b dsu (Ancestor Unit) = Ancestor t4
i: nat
j: i < length dsu
als: noIllegalIndices (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
hL: 2 ≤ length dsu
hs: ancestor dsu (length dsu) i ≠ a
ht: ancestor dsu (length dsu) i ≠ b
ter: validChainToAncestor (<[a:=ReferTo b]> (<[b:=Ancestor (Unite t2 t1)]> dsu))
(ancestorChain dsu (length dsu) i)
nn: ancestorChain dsu (length dsu) i =
ancestorChain
  (<[a:=ReferTo b]>
     (<[b:=Ancestor (Unite t2 t1)]> dsu))
  (length
     (<[a:=ReferTo b]>
        (<[b:=Ancestor (Unite t2 t1)]> dsu))) i
match
  nth (ancestor dsu (length dsu) i) dsu
    (Ancestor Unit)
with
| ReferTo _ => false
| Ancestor _ => true
end apply (h1 i j).
Qed.

Lemma unitePreservesWithoutCycles (dsu : list Slot) (a b : nat) (h : noIllegalIndices dsu) (h1 : withoutCyclesN dsu (length dsu)) : withoutCyclesN (unite dsu a b) (length dsu).
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
withoutCyclesN (unite dsu a b) (length dsu)
Proof.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
withoutCyclesN (unite dsu a b) (length dsu)
  unfold unite.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
withoutCyclesN
  (if decide (length dsu ≤ a)
   then dsu
   else
    if decide (length dsu ≤ b)
    then dsu
    else
     if
      decide
        (ancestor dsu (length dsu) a =
         ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
     then
      pathCompress
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
     else
      match
        nth (ancestor dsu (length dsu) a)
          (pathCompress
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _0 =>
          match
            nth
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
              (pathCompress
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b)) (Ancestor Unit)
          with
          | ReferTo _ =>
              pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
          | Ancestor _1 =>
              if decide (leafCount _1 < leafCount _0)
              then
               performMerge
                 (pathCompress
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b
                    (ancestor
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a))
                       (length
                          (pathCompress dsu
                             (length dsu) a
                             (ancestor dsu
                                (length dsu) a))) b))
                 _1 _0
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b) (ancestor dsu (length dsu) a)
              else
               performMerge
                 (pathCompress
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b
                    (ancestor
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a))
                       (length
                          (pathCompress dsu
                             (length dsu) a
                             (ancestor dsu
                                (length dsu) a))) b))
                 _0 _1 (ancestor dsu (length dsu) a)
                 (ancestor
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a))
                    (length
                       (pathCompress dsu (length dsu)
                          a
                          (ancestor dsu (length dsu) a)))
                    b)
          end
      end) (length dsu) case_decide as a1.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: length dsu ≤ a
withoutCyclesN dsu (length dsu)
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
withoutCyclesN
  (if decide (length dsu ≤ b)
   then dsu
   else
    if
     decide
       (ancestor dsu (length dsu) a =
        ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    then
     pathCompress
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))) b
       (ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    else
     match
       nth (ancestor dsu (length dsu) a)
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         (Ancestor Unit)
     with
     | ReferTo _ =>
         pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)
     | Ancestor _0 =>
         match
           nth
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) (Ancestor Unit)
         with
         | ReferTo _ =>
             pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
         | Ancestor _1 =>
             if decide (leafCount _1 < leafCount _0)
             then
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _1 _0
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b) (ancestor dsu (length dsu) a)
             else
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _0 _1
                (ancestor dsu (length dsu) a)
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
         end
     end) (length dsu) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: length dsu ≤ a
withoutCyclesN dsu (length dsu) exact h1. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
withoutCyclesN
  (if decide (length dsu ≤ b)
   then dsu
   else
    if
     decide
       (ancestor dsu (length dsu) a =
        ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    then
     pathCompress
       (pathCompress dsu (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))) b
       (ancestor
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b)
    else
     match
       nth (ancestor dsu (length dsu) a)
         (pathCompress
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         (Ancestor Unit)
     with
     | ReferTo _ =>
         pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b)
     | Ancestor _0 =>
         match
           nth
             (ancestor
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
             (pathCompress
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)) (Ancestor Unit)
         with
         | ReferTo _ =>
             pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
         | Ancestor _1 =>
             if decide (leafCount _1 < leafCount _0)
             then
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _1 _0
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b) (ancestor dsu (length dsu) a)
             else
              performMerge
                (pathCompress
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b
                   (ancestor
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a))
                      (length
                         (pathCompress dsu
                            (length dsu) a
                            (ancestor dsu (length dsu)
                               a))) b)) _0 _1
                (ancestor dsu (length dsu) a)
                (ancestor
                   (pathCompress dsu (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu (length dsu) a
                         (ancestor dsu (length dsu) a)))
                   b)
         end
     end) (length dsu)
  case_decide as a2.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: length dsu ≤ b
withoutCyclesN dsu (length dsu)
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
withoutCyclesN
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu (length dsu) a
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _1 _0
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
        end
    end) (length dsu) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: length dsu ≤ b
withoutCyclesN dsu (length dsu) exact h1. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
withoutCyclesN
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu (length dsu) a
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _1 _0
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
        end
    end) (length dsu)
  assert (hR : withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
     (length (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)))
     b
     (ancestor
        (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))) b))
  (length dsu)).
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
withoutCyclesN
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu (length dsu) a
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _1 _0
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a))
                     (length
                        (pathCompress dsu (length dsu)
                           a
                           (ancestor dsu (length dsu)
                              a))) b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu (length dsu) a
                        (ancestor dsu (length dsu) a)))
                  b)
        end
    end) (length dsu)
  {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu) pose proof pathCompressPreservesWithoutCycles _ h1 h (length dsu) a ltac:(lia) as step1.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
step1: withoutCyclesN
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length dsu)
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu) rewrite !pathCompressPreservesLength.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
step1: withoutCyclesN
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length dsu)
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a)) 
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)) 
        (length dsu) b)) 
  (length dsu)
    pose proof pathCompressPreservesWithoutCycles (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)) ltac:(rewrite pathCompressPreservesLength; exact step1) ltac:(apply pathCompressPreservesNoIllegalIndices; try rewrite pathCompressPreservesLength; try (assumption || lia); apply ancestorLtLength; (assumption || lia)) (length dsu) b ltac:(rewrite pathCompressPreservesLength; lia) as step2.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
step1: withoutCyclesN
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length dsu)
step2: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a)))
        b))
  (length
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a)))
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a)) 
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)) 
        (length dsu) b)) 
  (length dsu) rewrite !pathCompressPreservesLength in step2.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
step1: withoutCyclesN
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length dsu)
step2: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length dsu) b)) 
  (length dsu)
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a)) 
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)) 
        (length dsu) b)) 
  (length dsu) exact step2. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
withoutCyclesN
  (if
    decide
      (ancestor dsu (length dsu) a =
       ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   then
    pathCompress
      (pathCompress dsu 
         (length dsu) a 
         (ancestor dsu (length dsu) a))
      (length
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))) b
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
   else
    match
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b))
        (Ancestor Unit)
    with
    | ReferTo _ =>
        pathCompress
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b
          (ancestor
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
    | Ancestor _0 =>
        match
          nth
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
            (pathCompress
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b
               (ancestor
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                  b)) (Ancestor Unit)
        with
        | ReferTo _ =>
            pathCompress
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))) b
              (ancestor
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a))
                 (length
                    (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                 b)
        | Ancestor _1 =>
            if decide (leafCount _1 < leafCount _0)
            then
             performMerge
               (pathCompress
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a))
                     (length
                       (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                     b)) _1 _0
               (ancestor
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                  b) (ancestor dsu (length dsu) a)
            else
             performMerge
               (pathCompress
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                  b
                  (ancestor
                     (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a))
                     (length
                       (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                     b)) _0 _1
               (ancestor dsu (length dsu) a)
               (ancestor
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))
                  (length
                     (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                  b)
        end
    end) (length dsu)
  case_decide as a3.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))) b
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
withoutCyclesN
  match
    nth (ancestor dsu (length dsu) a)
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end (length dsu) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a =
ancestor
  (pathCompress dsu (length dsu) a
     (ancestor dsu (length dsu) a))
  (length
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))) b
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu) exact hR. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
withoutCyclesN
  match
    nth (ancestor dsu (length dsu) a)
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end (length dsu)
  remember (nth _ _ _) as g eqn:hg.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Slot
hg: g =
nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
withoutCyclesN
  match g with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end (length dsu) symmetry in hg.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Slot
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = g
withoutCyclesN
  match g with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      match
        nth
          (ancestor
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b)
          (pathCompress
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))
             (length
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))) b
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)) (Ancestor Unit)
      with
      | ReferTo _ =>
          pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b)
      | Ancestor _1 =>
          if decide (leafCount _1 < leafCount _0)
          then
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                   b)) _1 _0
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b) (ancestor dsu (length dsu) a)
          else
           performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a))
                   (length
                      (pathCompress dsu 
                       (length dsu) a
                       (ancestor dsu (length dsu) a)))
                   b)) _0 _1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu (length dsu) a)))
                b)
      end
  end (length dsu) destruct g as [g | g].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: nat
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo g
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
withoutCyclesN
  match
    nth
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount g)
      then
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 g
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         g _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end (length dsu) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: nat
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo g
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu) exact hR. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
withoutCyclesN
  match
    nth
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      (Ancestor Unit)
  with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount g)
      then
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 g
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         g _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end (length dsu)
  remember (nth (ancestor _ _ b) _ _) as g1 eqn:hg1.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: Slot
hg1: g1 =
nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit)
withoutCyclesN
  match g1 with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount g)
      then
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 g
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         g _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end (length dsu) symmetry in hg1.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: Slot
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = g1
withoutCyclesN
  match g1 with
  | ReferTo _ =>
      pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)
  | Ancestor _0 =>
      if decide (leafCount _0 < leafCount g)
      then
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         _0 g
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
         (ancestor dsu (length dsu) a)
      else
       performMerge
         (pathCompress
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b
            (ancestor
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))
               (length
                  (pathCompress dsu 
                     (length dsu) a
                     (ancestor dsu (length dsu) a))) b))
         g _0 (ancestor dsu (length dsu) a)
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b)
  end (length dsu) destruct g1 as [g1 | g1].
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: nat
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo g1
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: Tree
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g1
withoutCyclesN
  (if decide (leafCount g1 < leafCount g)
   then
    performMerge
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      g1 g
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (ancestor dsu (length dsu) a)
   else
    performMerge
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      g g1 (ancestor dsu (length dsu) a)
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b))
  (length dsu) {
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: nat
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = ReferTo g1
withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu) exact hR. }
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: Tree
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g1
withoutCyclesN
  (if decide (leafCount g1 < leafCount g)
   then
    performMerge
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      g1 g
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b)
      (ancestor dsu (length dsu) a)
   else
    performMerge
      (pathCompress
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b
         (ancestor
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))
            (length
               (pathCompress dsu 
                  (length dsu) a
                  (ancestor dsu (length dsu) a))) b))
      g g1 (ancestor dsu (length dsu) a)
      (ancestor
         (pathCompress dsu 
            (length dsu) a
            (ancestor dsu (length dsu) a))
         (length
            (pathCompress dsu 
               (length dsu) a
               (ancestor dsu (length dsu) a))) b))
  (length dsu)
  case_decide as hs.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: Tree
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g1
hs: leafCount g1 < leafCount g
withoutCyclesN
  (performMerge
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
     g1 g
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)
     (ancestor dsu (length dsu) a)) 
  (length dsu)
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: Tree
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g1
hs: ¬ leafCount g1 < leafCount g
withoutCyclesN
  (performMerge
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)) g
     g1 (ancestor dsu (length dsu) a)
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
  -
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: Tree
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g1
hs: leafCount g1 < leafCount g
withoutCyclesN
  (performMerge
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
     g1 g
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)
     (ancestor dsu (length dsu) a)) 
  (length dsu) pose proof performMergePreservesWithoutCycles (pathCompress
        (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)))
           b)) (ancestor
        (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))) b)
     (ancestor dsu (length dsu) a) ltac:(rewrite !pathCompressPreservesLength,!pathCompressPreservesAncestorLength; try (assumption || lia); apply ancestorLtLength; (assumption || lia)) ltac:(rewrite !pathCompressPreservesLength; apply ancestorLtLength; (assumption || lia)) ltac:(rewrite !pathCompressPreservesLength,!pathCompressPreservesAncestorLength in *; try (assumption || lia)) g1 g ltac:(apply pathCompressPreservesNoIllegalIndices; (try apply pathCompressPreservesNoIllegalIndices); (try rewrite !pathCompressPreservesLength); (try lia); (try assumption); [| rewrite pathCompressPreservesAncestorLength; try (assumption || lia)]; apply ancestorLtLength; (assumption || lia)) ltac:(pose proof pathCompressPreservesWithoutCycles (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)) as kw; rewrite !pathCompressPreservesLength in kw; rewrite !pathCompressPreservesLength; apply kw; (try lia); (try apply pathCompressPreservesNoIllegalIndices); (try apply pathCompressPreservesWithoutCycles); (try assumption); (try lia); apply ancestorLtLength; (assumption || lia)) as ii.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: Tree
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g1
hs: leafCount g1 < leafCount g
ii: ∀ _0 : Tree,
  nth
    (ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b)
    (pathCompress
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b))
    (Ancestor Unit) = Ancestor _0
  → ∀ _1 : Tree,
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a)))
           b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)) (Ancestor Unit) = 
      Ancestor _1
      → withoutCyclesN
          (performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))) b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))
                   (length
                      (pathCompress dsu
                       (length dsu) a
                       (ancestor dsu 
                       (length dsu) a))) b)) g1 g
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))) b)
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))) b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))
                   (length
                      (pathCompress dsu
                       (length dsu) a
                       (ancestor dsu 
                       (length dsu) a))) b)))
withoutCyclesN
  (performMerge
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b))
     g1 g
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b)
     (ancestor dsu (length dsu) a)) 
  (length dsu)
     rewrite !pathCompressPreservesLength in *.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length dsu) b)) 
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length dsu) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length dsu) b)) 
  (Ancestor Unit) = Ancestor g
g1: Tree
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length dsu) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length dsu) b)) 
  (Ancestor Unit) = Ancestor g1
hs: leafCount g1 < leafCount g
ii: ∀ _0 : Tree,
  nth
    (ancestor
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length dsu) b)
    (pathCompress
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length dsu) b
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length dsu) b)) 
    (Ancestor Unit) = Ancestor _0
  → ∀ _1 : Tree,
      nth (ancestor dsu (length dsu) a)
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length dsu) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length dsu) b)) 
        (Ancestor Unit) = 
      Ancestor _1
      → withoutCyclesN
          (performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length dsu) b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))
                   (length dsu) b)) g1 g
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length dsu) b)
             (ancestor dsu (length dsu) a))
          (length dsu)
withoutCyclesN
  (performMerge
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)) 
        (length dsu) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length dsu) b)) g1 g
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)) 
        (length dsu) b) 
     (ancestor dsu (length dsu) a)) 
  (length dsu)
     exact (ii g1 hg1 g hg).
  -
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: Tree
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g1
hs: ¬ leafCount g1 < leafCount g
withoutCyclesN
  (performMerge
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)) g
     g1 (ancestor dsu (length dsu) a)
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu) pose proof performMergePreservesWithoutCycles (pathCompress
        (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)))
           b)) (ancestor dsu (length dsu) a)
     (ancestor
        (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a))) b) ltac:(rewrite !pathCompressPreservesLength; apply ancestorLtLength; (assumption || lia)) ltac:(rewrite !pathCompressPreservesLength,!pathCompressPreservesAncestorLength; try (assumption || lia); apply ancestorLtLength; (assumption || lia)) ltac:(rewrite !pathCompressPreservesLength,!pathCompressPreservesAncestorLength in *; try (assumption || lia)) g g1 ltac:(apply pathCompressPreservesNoIllegalIndices; (try apply pathCompressPreservesNoIllegalIndices); (try rewrite !pathCompressPreservesLength); (try lia); (try assumption); [| rewrite pathCompressPreservesAncestorLength; try (assumption || lia)]; apply ancestorLtLength; (assumption || lia)) ltac:(pose proof pathCompressPreservesWithoutCycles (pathCompress dsu (length dsu) a (ancestor dsu (length dsu) a)) as kw; rewrite !pathCompressPreservesLength in kw; rewrite !pathCompressPreservesLength; apply kw; (try lia); (try apply pathCompressPreservesNoIllegalIndices); (try apply pathCompressPreservesWithoutCycles); (try assumption); (try lia); apply ancestorLtLength; (assumption || lia)) as ii.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g
g1: Tree
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (Ancestor Unit) = Ancestor g1
hs: ¬ leafCount g1 < leafCount g
ii: ∀ _0 : Tree,
  nth (ancestor dsu (length dsu) a)
    (pathCompress
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))) b
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length
             (pathCompress dsu 
                (length dsu) a
                (ancestor dsu (length dsu) a))) b))
    (Ancestor Unit) = Ancestor _0
  → ∀ _1 : Tree,
      nth
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a)))
           b)
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a)))
           b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length
                 (pathCompress dsu 
                    (length dsu) a
                    (ancestor dsu (length dsu) a)))
              b)) (Ancestor Unit) = 
      Ancestor _1
      → withoutCyclesN
          (performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))) b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))
                   (length
                      (pathCompress dsu
                       (length dsu) a
                       (ancestor dsu 
                       (length dsu) a))) b)) g g1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))) b))
          (length
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))) b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))
                   (length
                      (pathCompress dsu
                       (length dsu) a
                       (ancestor dsu 
                       (length dsu) a))) b)))
withoutCyclesN
  (performMerge
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))) b)) g
     g1 (ancestor dsu (length dsu) a)
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))) b))
  (length dsu)
     rewrite !pathCompressPreservesLength in *.
dsu: list Slot
a, b: nat
h: noIllegalIndices dsu
h1: withoutCyclesN dsu (length dsu)
a1: ¬ length dsu ≤ a
a2: ¬ length dsu ≤ b
hR: withoutCyclesN
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length dsu) b)) 
  (length dsu)
a3: ancestor dsu (length dsu) a
≠ ancestor
    (pathCompress dsu (length dsu) a
       (ancestor dsu (length dsu) a))
    (length dsu) b
g: Tree
hg: nth (ancestor dsu (length dsu) a)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length dsu) b)) 
  (Ancestor Unit) = Ancestor g
g1: Tree
hg1: nth
  (ancestor
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length dsu) b)
  (pathCompress
     (pathCompress dsu (length dsu) a
        (ancestor dsu (length dsu) a))
     (length dsu) b
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a))
        (length dsu) b)) 
  (Ancestor Unit) = Ancestor g1
hs: ¬ leafCount g1 < leafCount g
ii: ∀ _0 : Tree,
  nth (ancestor dsu (length dsu) a)
    (pathCompress
       (pathCompress dsu 
          (length dsu) a
          (ancestor dsu (length dsu) a))
       (length dsu) b
       (ancestor
          (pathCompress dsu 
             (length dsu) a
             (ancestor dsu (length dsu) a))
          (length dsu) b)) 
    (Ancestor Unit) = Ancestor _0
  → ∀ _1 : Tree,
      nth
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length dsu) b)
        (pathCompress
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length dsu) b
           (ancestor
              (pathCompress dsu 
                 (length dsu) a
                 (ancestor dsu (length dsu) a))
              (length dsu) b)) 
        (Ancestor Unit) = 
      Ancestor _1
      → withoutCyclesN
          (performMerge
             (pathCompress
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length dsu) b
                (ancestor
                   (pathCompress dsu 
                      (length dsu) a
                      (ancestor dsu 
                       (length dsu) a))
                   (length dsu) b)) g g1
             (ancestor dsu (length dsu) a)
             (ancestor
                (pathCompress dsu 
                   (length dsu) a
                   (ancestor dsu (length dsu) a))
                (length dsu) b)) 
          (length dsu)
withoutCyclesN
  (performMerge
     (pathCompress
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)) 
        (length dsu) b
        (ancestor
           (pathCompress dsu 
              (length dsu) a
              (ancestor dsu (length dsu) a))
           (length dsu) b)) g g1
     (ancestor dsu (length dsu) a)
     (ancestor
        (pathCompress dsu 
           (length dsu) a
           (ancestor dsu (length dsu) a)) 
        (length dsu) b)) 
  (length dsu)
     exact (ii g hg g1 hg1).
Qed.